UNIVERSITY  OF  CALIFORNIA 


c 

c« 


A   LABORATORY   MANUAL 


PHYSICS  AND  APPLIED    ELECTRICITY 


A  LABORATORY   MANUAL 


OF 


PHYSICS  AND  APPLIED  ELECTRICITY 


ARRANGED  AND   EDITED 
BY 

EDWARD    L..  NICHOLS 

.  *    7SV 

PROFESSOR  OF  PHYSiqg  ^N  CORMELL  UNIVERSITY 


IN"  TWO    VOLUMES 

VOL.  I 
JUNIOR  COURSE   IN   GENERAL   PHYSICS 

BY 

ERNEST  MERRITT  AND   FREDERICK  J.   ROGERS 


MACMILLAN    AND    CO. 

AND     LONDON 
1894 

All  rights  reserved 


IGHT,  1 
BY  MACMILLAN  AND  CO. 


Set  up  and  electrotyped  June,  1894.       Reprinted 
October,  1894. 


Norfooolf 

J.  S.  Gushing  &  Co.  — Berwick  &  Smith. 
Boston,  Mass.,  U.S.A. 


PREFACE. 


THIS  work  has  been  written  to  supply  in  some  measure  the  needs  of 
a  modern  laboratory,  in  which  the  existing  manuals  of  physics  have  been 
found  inadequate.  In  its  present  form  the  book  is  the  work,  chiefly, 
of  Assistant  Professors  George  S.  Moler,  Ernest  Merritt,  and  Frederick 
Bedell,  of  Instructors  Frederick  J.  Rogers,  Homer  J.  Hotchkiss,  Charles 
P.  Matthews,  and  of  the  editor.  Certain  parts,  however,  have  been 
taken  from  written  directions  to  students  which  had  been  prepared  by 
instructors  who  are  no  longer  members  of  the  department  from  which 
the  book  emanates,  and  who  have  taken  no  immediate  hand  in  its  final 
preparation. 

No  attempt  has  been  made  to  provide  a  complete  and  sufficient 
source  of  information  for  laboratory  students.  On  the  contrary,  it 
has  been  thought  wise  to  encourage  continual  reference  to  other 
works  and  to  original  sources.  It  is  assumed  that  in  all  laboratories 
in  which  a  work  of  this  kind  will  be  found  useful,  there  is  accessible 
to  the  student  a  small  collection  of  reference  volumes,  including  the 
Laboratory  Manuals  of  Kohlrausch,  Glazebrook  and  Shaw,  Stewart  and 
Gee,  Witz,  and  of  Wiedemann  and  Ebert ;  also  that  the  larger  treatises 
on  experimental  physics  of  Jamin,  Winkelmann,  Violle,  Wiedemann, 
Preston,  etc.,  together  with  the  best  known  of  the  lesser  works  in 
English,  are  available. 

The  Manual  has  been  divided  into  two  volumes ;  and  it  is  designed 
for  three  classes  of  students,  differing  from  each  other  in  experience, 
maturity,  and  purpose.  The  method  of  treatment  has  been  varied  in 
accordance  with  the  principle,  that  with  increasing  experience  the 
student  should  be  divorced  more  and  more  from  the  use  of  the  Manual 


vi  PREFACE. 

and  also  from  the  close  supervision  of  the  instructor,  and  that  he  should 
be  thrown  gradually  upon  his  own  resources,  and  be  led  to  make  a 
wider  and  wider  use  of  the  literature  of  the  science. 

It  will  be  found  that  the  first  volume,  which  is  intended  for  beginners, 
affords  explicit  directions,  together  with  demonstrations  and  occasional 
elementary  statements  of  principles.  This  volume  is  the  outgrowth  of  a 
system  of  junior  instruction  which  has  been  gradually  developed  during 
a  quarter  of  a  century.  No  attempt  has  been  made  to  include  the 
whole  of  physics.  On  the  other  hand,  the  principle  has  been  followed 
here,  as  indeed  throughout  the  book,  of  incorporating  only  such  experi- 
ments as  have  been  in  actual  use. 

It  is  assumed  that  the  student  possesses  some  knowledge  of  analyti- 
cal geometry  and  of  the  calculus ;  also  that  he  has  completed  a  text- 
book and  lecture  course  in  the  principles  of  physics.  It  is  not  expected 
that  the  experiments  will  be  taken  consecutively,  nor  that  a  student,  in 
the  time  usually  given  to  the  work,  will  complete  more  than  a  third 
of  them.  The  experiments  have  been  divided  into  groups,  an  arrange- 
ment of  the  work  for  which  there  were  two  reasons.  On  the  one 
hand,  it  serves  to  guide  the  practicant  and  the  instructor  in  the  selection 
of  experiments ;  on  the  other  hand,  it  furthers  the  development  of  the 
system  by  making  it  easy  to  add  or  to  exclude  material.  It  is  expected, 
indeed,  that  the  book  will  be  used  thus  by  those  into  whose  hands  it 
may  come,  each  one  adding  such  experiments  to  the  various  groups  as 
he  may  desire  to  include  in  his  course,  and  dropping  out  those  which 
he  may  deem  useless. 

In  the  second  volume  more  is  left  to  the  individual  effort  and  to  the 
maturer  intelligence  of  the  practicant.  This  volume  differs  from  the  first 
also  in  another  respect.  In  the  junior  course  no  attempt  is  made  to 
leave  the  beaten  track.  The  very  nature  of  the  subjects  with  which  we 
have  to  deal  in  Volume  II,  however,  has  compelled  the  introduction  of 
new  materials.  The  writers  trust  that  where  the  ripeness  and  maturity 
of  treatment  which  comes  from  long-continued  experience  in  the  teach- 
ing of  a  subject  is  missing,  some  recompense  may  be  found  in  the 
freshness  and  novelty  of  the  themes. 


PREFACE.  vii 

A  large  proportion  of  the  students,  for  whom  primarily  this  Manual 
is  intended,  are  preparing  to  become  engineers,  and  especial  attention 
has  been  devoted  to  the  needs  of  that  class  of  readers.  In  Parts  I,  II, 
and  III  of  Volume  II,  especially,  a  considerable  amount  of  work  in 
applied  electricity,  in  photometry,  and  in  heat  has  been  introduced, 
with  particular  reference  to  the  training  of  students  of  engineering.  It 
is  believed,  nevertheless,  that  selections  from  these  parts  may  be  made 
which  will  be  of  value  to  students  of  pure  physics  also. 

The  final  chapters  (Part  IV),  which  are  intended  for  those  who 
have  already  had  two  years  or  more  of  laboratory  instruction,  consist 
simply  of  certain  hints  for  advanced  work.  These  are  accompanied  by 
typical  results,  the  object  of  which  is  to  show  in  brief  form  what  has 
already  been  accomplished  by  the  methods  proposed,  and  to  lead  the 
student  to  a  suitable  starting-point  for  further  investigation.  Through- 
out this  portion  of  the  book  theory  and  experimental  detail  alike  have 
been  omitted.  The  outlines  which  have  been  given  are  designed  to 
afford  suggestions  only,  and  by  virtue  of  their  very  meagreness  to  com- 
pel the  student  to  read  original  memoirs  in  preparation  for  his  work. 

EDWARD  L.  NICHOLS. 

CORNELL  UNIVERSITY,  ITHACA,  NEW  YORK, 
May,  1894. 


TJNIVEI,  :IT7 


•*OI  THB  = 


-f 

TABLE   OF   CONTENTS. 


VOLUME    I. 

PAGE 

INTRODUCTION i 

Record  of  Observations.  Units.  Graphical  Representation  of  Results. 
Errors  of  Observation  and  Method  of  Least  Squares. 

CHAPTER  I 26 

Curvature  of  a  Lens  by  the  Spherometer.  Adjustment  of  a  Cathetometer. 
Calibration  of  a  Thermometer  Tube.  Volume  and  Density  by  Measure- 
ment. Periodic  Motion  by  Method  of  Middle  Elongations.  Parallelo- 
gram of  Forces.  Parallel  Forces.  Principle  of  Moments.  Coefficient 
of  Friction.  Wheel  and  Axle.  Efficiency  of  a  System  of  Pulleys. 
Atwood's  Machine.  Gravity  from  Motion  of  a  Freely  Falling  Body. 
General  Statements  concerning  Moment  of  Inertia  and  Simple  Har- 
monic Motion.  Gravity  by  the  Physical  Pendulum.  Gravity  by  Rater's 
Pendulum.  Variation  of  Period  of  Bar  Pendulum  with  Position  of  Knife 
Edges.  Moment  of  Inertia.  Young's  Modulus  by  Stretching.  Moment 
of  Torsion.  Moment  of  Inertia  by  Torsion. 

CHAPTER  II 79 

General  Statements  concerning  Density.  Approximate  Determination  of 
Density  by  weighing  in  Water.  Specific  Gravity  Bottle.  Density  with 
Corrections  for  Temperature  and  Air  Displacement.  Jolly  Balance. 
Nicholson's  Hydrometer.  Fahrenheit's  Hydrometer.  Graduation  of  a 
Hydrometer.  Density  of  a  Solid  by  Variable  Immersion  Hydrometers. 
Hare's  Method  of  determining  Density  of  a  Liquid.  Verification  of 
Boyle's  Law.  Comparison  of  Barometers.  Expansion  of  Air. 

CHAPTER  III ioj 

General  Statements  concerning  Calorimetry.  Heat  of  Vaporization  of 
Water.  Heat  of  Fusion  of  Ice.  Specific  Heat  of  a  Solid.  Radiating 
and  Absorbing  Powers  of  Surfaces. 

CHAPTER  IV 122 

General  Statements  concerning  Static  Electricity.  Electrostatic  Induc- 
tion. The  Principle  of  the  Condenser.  The  Holtz  Machine.  The 
Holtz  Machine  (continued). 

CHAPTER  V 138 

General  Statements  concerning  Magnetism.  Lines  of  Force  and  Study  of 
Magnetic  Fields.  Magnetic  Moment  by  Method  of  Oscillations.  Mag- 
netic Moment  by  the  Magnetometer.  Measurement  of  the  Intensity  of 
a  Magnetic  Field.  Distribution  of  "  Free  "  Magnetism  in  a  Permanent 
Ma.gnet. 


x  TABLE   OF   CONTENTS. 

PAGE 

CHAPTER  Vl 153 

General  Statements  concerning  the  Electric  Current.  Law  of  the  Tangent 
Galvanometer.  Measurement  of  Current  by  Electrolysis.  Measurement 
of  the  Constant  of  a  Sensitive  Galvanometer.  Theory  of  Shunts.  Appli- 
cations of  the  Galvanometer  to  the  Measurement  of  Current. 

CHAPTER  VII 181 

General  Statements  concerning  Difference  of  Potential  and  Electromotive 
Force.  Comparison  of  two  Electromotive  Forces.  Ohm's  Method  for 
the  Measurement  of  E.  M.  F.  Potential  Difference  <at  the  Terminals  of 
a  Battery.  Fall  of  Potential  in  a  Wire  carrying  a  Current.  Beetz's 
Method  of  measuring  E. -M.  F.  To  trace  the  Lines  of  Equal  Potential 
in  a  Liquid  Conductor.  Variation  of  the  E.  M.  F.  of  a  Thermo-element. 

CHAPTER  VIII 204 

General  Statements  concerning  Resistance.  Measurement  of  Resistance 
by  the  Wheatstone  Bridge.  Measurement  of  Resistance  by  the  Fall  of 
Potential  Method.  Measurement  of  Specific  Resistance.  Temperature 
Coefficient  for  Resistance.  Internal  Resistance  of  a  Battery  by  Ohm's 
Method.  Resistance  of  a  Battery  by  Mance's  Method.  Resistance  of 
Electrolytes. 

CHAPTER  IX 220 

General  Statements  concerning  Electrical  Quantity.  Constant  of  a  Ballistic 
Galvanometer.  Logarithmic  Decrement  of  a  Galvanometer  Needle. 
Comparison  of  the  Capacities  of  Two  Condensers.  Measurement  of 
Capacity  in  Absolute  Measure. 

CHAPTER  X 232 

General  Statements  concerning  Induction.  Dip  and  Intensity  of  the 
Earth's  Magnetic  Field.  (Method  of  the  Earth's  Inductor.)  Measure- 
ment of  the  Lines  of  Force  of  a  Permanent  Magnet.  Mutual  Induction. 

CHAPTER  XI 245 

Measurement  of  Pitch  by  the  Syren.  Interference  and  Measurement  of 
Wave  length  by  Koenig's  Apparatus.  Resonance  of  Columns  of  Air  and 
the  Velocity  of  Sound.  Velocity  of  Sound  in  Brass.  The  Sonometer. 
Study  of  Transverse  Vibrations  by  Moler's  Method. 

CHAPTER  XII 257 

Radius  of  Curvature  by  Reflection.  Focal  Length  of  a  Concave  Mirror. 
Focal  Length  of  a  Convex  Lens.  Magnifying  Power  of  a  Telescope. 
Magnifying  Power  of  a  Microscope,  i  Measurement  of  the  Index  of  Re- 
fraction. >  Study  of  Flame  Spectra  of  Various  Metals.  Determination  of 
the  Distance  between  the  Lines  of  a  Grating.  Measurement  of  Candle 
Power  by  the  Bunsen  Photometer. 


.  ^"^^^5^. 

TABLE   OF   CONTENTS.       tfls*  xi 

frar/fe*  ^ 

&*+  y 

VOLUME   II. 
PART  I. 

EXPERIMENTS    WITH  DIRECT  CURRENT  APPARATUS. 

GENERAL  INTRODUCTION 

Study  of  a  Dynamo.  Introductory  to  Characteristics  of  Dynamos.  Char- 
acteristics of  a  Series  Dynamo.  Characteristics  of  a  Shunt  Dynamo. 
Armature  Characteristic.  Characteristics  of  a  Compound  Dynamo. 
Comparison  of  Magnetization  Curves  of  Dynamos.  Characteristics  of 
the  Waterhouse  Dynamo  and  Study  of  Third  Brush  Regulation.  Charac- 
teristics of  Edison  Arc  Dynamo.  Characteristics  of  Thomson-Houston 
Arc  Dynamo.  Characteristics  of  the  Ball  Dynamo.  Study  of  Brackett- 
Cradle  Dynamometer.  Efficiency  of  Double  Transformation  of  a  Small 
Motor  and  Dynamo.  Efficiency  of  a  Small  Dynamo.  Efficiency  of  a 
Small  Motor  by  Use  of  a  Rafford  Dynamometer.  Efficiency  of  a  Motor 
without  a  Dynamometer.  Efficiency  of  a  Dynamo  without  a  Dynamom- 
eter. Efficiency  of  a  Motor  with  a  Cradle  Dynamometer.  Efficiency 
of  a  Dynamo  with  a  Cradle  Dynamometer.  Reversing  Motor.  Study 
of  Arc-lamps.  Determination  of  the  Constants  of  a  Tangent  Galva- 
nometer from  its  Dimensions.  Use  of  Great  Tangent  Galvanometer  and 
Verification  of  Constants  by  Experiment.  Conditions  of  Maximum  Sensi- 
tiveness for  Great  Tangent  Galvanometer.  Determination  of  H  by  the 
Tangent  Galvanometer  Method.  Computation  of  the  Resistance  neces- 
sary to  render  a  Potential  Galvanometer  Direct  Reading.  Determina- 
tion of  Magnetic  Dip.  Introductory  to  the  Calibration  of  Instruments. 
Calibrating  an  Ammeter;  Voltmeter;  Electrodynamometer.  Constants 
of  Graded  Ammeter;  Voltmeter.  Reliability  Test  of  a  Voltmeter;  Am- 
meter. Exploration  of  the  Field  of  a  Dynamo.  Exploration  Curves  by 
Means  of  a  Dynamo  Indicator.  Determination  of  the  Coefficient  of 
Magnetic  Leakage  in  a  Dynamo.  Distribution  of  Waste  Magnetic  Flux 
in  a  Dynamo.  Measurement  of  Resistance  by  the  Ammeter  and  Volt- 
meter Method.  Study  of  a  Resistance.  Construction  and  Test  of  Fuse 
Wires.  Adjustment  and  Test  of  Accumulators.  Loss  of  Electromotive 
Force  Due  to  Self-induction  in  a  Dynamo.  Compounding  a  Dynamo 
from  its  Armature  Characteristic.  Compounding  a  Dynamo  by  Added 
Turns.  Economic  Coefficient  of  a  Series  Dynamo.  Economic  Coeffi- 
cient of  a  Shunt  Dynamo.  Determination  of  Air  Gap;  of  Armature 
Turns;  of  Back  Turns;  of  Field  Turns.  Power  lost  in  an  Armature  Due 
to  Hysteresis  and  Foucault  Currents.  Separation  of  Losses  in  a  Dynamo. 
Speed  Curves  of  a  Series  Dynamo.  Speed  Curves  of  a  Shunt  Dynamo. 
Curve  of  Torque  of  a  Motor  by  running  it  as  a  Dynamo.  Mechanical 
Characteristic  of  a  Motor. 


xii  TABLE  OF   CONTENTS. 

PART  II. 

EXPERIMENTS  IN  ALTERNATING   CURRENTS. 

INTRODUCTION  TO  ALTERNATING  CURRENT  EXPERIMENTS 

Curve  of  Magnetization  of  Alternating  Current  Generator.  Study  of  Alter- 
nating-Current Generator.  External  Characteristic  of  Alternating  Current 
Generator.  Alternating  Current  Potentiometer;  Adjustment  and  Test  for 
Sensitiveness.  Curve  of  Magnetization  of  Alternating  Current  Generator : 
Ballistic  Method.  Exploring  Field  of  Alternator.  Measurement  of  the 
Coefficient  of  Self-induction :  Impedance  Method.  Measurement  of  the 
Coefficient  of  Self-induction :  Three  Voltmeter  Method.  Variation  in 
the  Coefficient  of  Self-induction  with  the  Current.  Variation  in  the  Co- 
efficient of  Self-induction  with  the  Saturation  of  the  Iron  Core.  Effects 
of  the  Variation  of  the  Resistance  in  a  Series  Circuit.  Effects  of  the 
Variation  of  the  Self-induction  in  a  Series  Circuit.  Electromotive  Forces 
in  a  Series  Circuit.  Measurement  of  Power :  Three  Voltmeter  Method. 
Measurement  of  Power :  Three  Ammeter  Method.  Equivalent  Resist- 
ance and  Self-induction  of  Parallel  Circuits.  Effect  of  Frequency  upon 
Impedance  of  a  Circuit  containing  Resistance  and  Self-induction.  Effect 
of  Frequency  upon  Angle  of  Lag  in  a  Circuit  containing  Resistance  and 
Self-induction.  Measurement  of  Mutual  Induction :  Ballistic  Method. 
Measurement  of  Mutual  Induction :  Alternating  Current  Method.  Study 
of  a  Transformer.  Transformer  Test :  Three  Voltmeter  Method  at  No 
Load.  Transformer  Test :  Three  Voltmeter  Method  at  All  Loads.  Trans- 
former Test :  Three  Ammeter  Method  at  No  Load.  At  All  Loads.  Trans- 
former Test:  Variations  in  Transformer  Diagrams.  Operation  of  a 
Synchronous  Motor.  Introductory  to  Experiments  with  Condensers. 
Study  of  Standard  Condenser.  Curves  of  Condenser  Discharge :  Ballistic 
Method.  Curves  of  Condenser  Discharge  :  Potential  Method.  Curves  of 
Condenser  Discharge:  Deflection  Method.  Measurement  of  Capacity 
by  Curves  of  Condenser  Discharge.  Measurement  of  Resistance  by  Curves 
W  Condenser  Discharge.  Comparison  of  Capacities :  Direct  Deflection 
Method.  Comparison  of  Capacities :  Method  of  Mixtures.  Comparison 
of  Capacities :  Gott's  Method.  Comparison  of  Capacities :  Bridge  Method. 
Comparison  of  Capacities:  Divided  Charge  Method.  Comparison  of 
Capacities :  Diminished  Charge  Method.  Capacities  in  Parallel  and 
Series.  Comparison  of  Electromotive  Forces  by  a  Condenser.  Study  of 
Residual  Discharges.  Measurement  of  Capacity  by  Alternating  Current 
Method.  Effects  of  the  Variation  of  the  Resistance  in  a  Series  Circuit 
containing  a  Condenser.  Effects  of  the  Variation  of  the  Capacity  in  a 
Series  Circuit.  Effects  of  the  Variation  of  Frequency  in  a  Circuit  contain- 
ing Capacity  but  No  Self-induction.  Neutralization  of  Self-induction 
and  Capacity  in  Series.  Self-induction  and  Capacity  in  Parallel.  Instan- 
taneous Measurement  with  a  Revolving  Contact  Maker  and  Electrostatic 
Voltmeter.  Instantaneous  Measurement  with  a  Revolving  Contact 
Maker :  Telephone  Method.  Instantaneous  Measurement  with  a  Revolv- 


TABLE   OF   CONTENTS.  xiii 


ing  Contact  Maker :  Ballistic  Method.  Irregularities  in  Alternating 
Current  Curves.  Measurement  of  Power  by  the  Method  of  Instantaneous 
Contact.  Transformer  Test  by  the  Method  of  Instantaneous  Contact. 
Study  of  the  Effects  of  Capacity  by  Method  of  Instantaneous  Contact. 
Determination  of  Dielectric  Hysteresis.  Test  of  a  Non-inductive  Resist- 
ance by  the  Method  of  Instantaneous  Contact.  Investigation  of  Liquid 
Resistance.  Calibration  of  a  Hot-wire  Voltmeter.  Calibration  of  a  Hot- 
wire Ammeter.  Determination  of  the  Constant  of  a  Ballistic  Galvanom- 
eter. Calibration  of  D'Arsonval  or  Ballistic  Galvanometer  for  Potential. 
Magnetic  Qualities  of  Iron :  Ring  Method.  Magnetic  Qualities  of  Iron : 
Instantaneous  Contact  Method.  Illustrative  Experiments  with  Alternat- 
ing Current  Magnet. 

PART    III. 

SENIOR   COURSES  IN  PHOTOMETRY  AND  HEAT. 

CHAPTER  I 

Standardization  of  Instruments.     Distribution  of  Candle-power  about  an 
Incandescent  Lamp.     Characteristic  Curves  of  an  Incandescent  Lamp. 
Photometry  of  the  Arc-light. 
CHAPTER  II 

Specific  Heat  by  the  Method  of  Mixtures.  Specific  Heat  by  the  Method  of 
Mixtures,  Liquids.  Specific  Heat  by  the  Method  of  Cooling.  Specific 
Heat  by  the  Bunsen  Ice  Calorimeter.  Use  of  Favre  and  Silbermann's 
Calorimeter.  Pressure  of  Saturated  Vapors  at  Low  Temperatures. 
Pressure  of  Saturated  Vapors  at  High  Temperatures.  Density  of  Vapors 
(Dumas'  Method).  Heat  of  Vaporization.  Heat  of  Combustion  of 
Metals.  Cubical  Expansion  of  Solids  by  the  Method  of  Balance  Transits. 
Mechanical  Equivalent  of  Heat  by  Current  Calorimeter.  Measurement 
of  Temperatures  by  the  Thermal  Element. 

PART  IV. 

OUTLINES   OF  ADVANCED    WORK. 

INTRODUCTION 

CHAPTER  I 

Critical  Study  of  Thermometers.     Influence  of  Temperature  upon  Young's 

Modulus.     Thermal  Conductivity  of  a  Copper  Bar.     Volume  of  Liquid 

Mixtures.     Volume  of  Substances  near  the  Melting-point.     Influence  of 

Temperature  upon  the  Color  of  Pigments.     Transparency  of  Solutions. 

CHAPTER  II 

Efficiency  of  Artificial  Light  Sources.     The  Glow-lamp.     The  Arc-lamp. 

The  Magnesium  Light.     The  Drummond  Light.     Other  Sources.- 
CHAPTER  III 

Life  Studies  of  Light  Sources.  Incandescent  Oxides.  The  Glow-lamp. 
Candles,  Oil,  and  Gas. 


xiv  TABLE   OF   CONTENTS. 


CHAPTER  IV 

Spectrophotometry.  The  Construction  of  Spectrophotometers.  Compari- 
son of  Artificial  Light  Sources.  Daylight.  Radiation  as  a  Function  of 
Temperature.  The  Study  of  Pigments  and  Solutions. 

CHAPTER  V 

Studies  of  the  Invisible  Spectrum.     The  Infra-red.     The  Ultra-violet. 

CHAPTER  VI 

Physiological  Optics.  Duration  of  Impressions.  Luminosity  and  Dichroic 
Vision.  The  Neutral  Zone  in  the  Spectrum  of  the  Color-blind.  Per- 
sonal Errors  in  Photometry.  The  Light  passing  through  Sectored  Disks 
in  Rapid  Rotation. 

CHAPTER  VII 

Explorations  of  the  Earth's  Magnetic  Field.  The  Method  of  the  Kew  Mag- 
netometer. The  Method  of  the  Tangent  Galvanometer.  Appendix. 


A  LABORATORY  MANUAL  OF  PHYSICS 
AND  APPLIED  ELECTRICITY. 


VOLUME    I. 

JUNIOR    COURSE  IN  GENERAL    PHYSICS. 
BY  ERNEST  MERRITT  AND  FREDERICK  J.  ROGERS. 

INTRODUCTION. 

THE  object  of  all  of  the  experiments  described  in  the  fol- 
lowing pages  is  twofold  :  (i)  to  illustrate,  and  therefore  impress 
more  thoroughly  on  the  mind,  the  principles  and  laws  which 
have  previously  been  taught  by  text-books  or  lectures ;  (2)  to 
familiarize  the  student  with  proper  methods  of  observation  and 
physical  experimentation.  These  two  aims  should  be  kept  in 
view  throughout  the  work  which  follows. 

GENERAL     DIRECTIONS. 

Before  beginning  any  experimental  work,  the  student  is 
advised  to  read  carefully  the  directions  for  the  experiment  that 
is  to  be  performed,  making  sure  that  the  object  of  the  experi- 
ment and  the  means  to  be  employed  in  accomplishing  this 
object  are  fully  understood.  If  the  experiment  involves  prin- 
ciples which  are  unfamiliar,  the  matter  should  be  looked  up  in 


2  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

some  reference  book  before  the  observations  are  begun.  If  this 
is  done,  the  significance  of  each  step  in  the  experimental  work 
will  be  appreciated,  and  the  experiment  will  therefore  be  more 
instructive.  The  likelihood  of  essential  observations  being 
omitted  is  also  less  when  the  bearing  of  each  observation  upon 
the  result  is  fully  understood. 

Record  of  Observations. — All  original  observations  should  be 
recorded  in  a  note-book  at  the  time  when  they  are  taken,  and 
should  be  preserved.  It  is  a  saving  in  the  end  to  devote  enough 
time  to  the  original  records  to  make  them  neat  and  clear,  and 
so  complete  as  to  enable  any  person  who  is  familiar  with  the 
experiment  to  understand  the  meaning  of  each  figure  recorded. 

In  all  cases,  it  is  the  original  observations  that  are  to  be 
recorded.  A  derived  result  should  in  no  case  be  recorded  as 
an  observation,  no  matter  how  simple  may  be  the  process  of 
derivation. 

For  example,  it  may  be  required  to  find  the  duration  of  a  certain 
phenomenon ;  let  us  say  that  it  begins  at  half-past  three  o'clock  and 
lasts  until  twenty-two  minutes  of  four ;  the  time  is  eight  minutes,  but 
this  is  a  derived  result  obtained  by  subtracting  3.30  from  3.38.  The 
actual  time  of  beginning  and  end  should  be  recorded,  and  the  sub- 
traction performed  afterward. 

The  uniform  observance  of  this  rule  will  save  annoyance 
from  simple  mistakes  due  to  carelessness  or  haste,  which  are 
frequently  made  even  by  the  best  observers,  and  which,  without 
the  original  observations,  it  would  be  impossible  to  correct. 

The  time  at  which  each  observation  is  taken  should  always 
be  recorded,  including  the  day,  hour,  and  minute. 

An  example  of  the  possible  usefulness  of  this  rule  might  occur  in 
the  case  of  observations  taken  with  a  sensitive  galvanometer.  Let  us 
suppose  that  it  is  found  on  working  up  the  data  that  the  results  of 
certain  observations  disagree  with  those  of  the  remainder.  The  very 
annoying  question  then  arises  whether  this  disagreement  represents  an 
actual  change  in  the  phenomena  observed,  or  whether  it  is  due  to  the 


INTRODUCTION.  3 

effect  on  the  galvanometer  of  some  outside  disturbance.  Knowing  the 
time  at  which  the  observations  were  taken,  it  will  be  possible  to  investi- 
gate the  matter,  and  if  it  is  found  that  there  was  a  change  in  the  mag- 
netic conditions  (due  to  varying  currents,  moving  iron,  or  other  causes) 
at  the  time  when  the  irregularities  were  observed,  then  these  observa- 
tions may  be  legitimately  discarded. 

Observations.  —  It  is  to  be  remembered  that  the  object  of 
scientific  observations  is  not  to  confirm  preconceived  theories, 
or  to  obtain  a  series  of  results  which  shall  arouse  admiration  on 
account  of  their  uniformity,  but  to  discover  the  truth  in  regard 
to  the  phenomenon  investigated,  no  matter  what  the  truth  may 
be.  It  is  of  the  greatest  importance,  therefore,  that  the  observer 
should  be  entirely  unprejudiced,  either  by  a  knowledge  of  the 
results  of  other  experimenters,  or  by  any  preconceived  notion 
as  to  what  the  results  should  be.  It  is  not  meant  by  this  that 
the  observer  must  be  ignorant  of  the  probable  results  :  but  that 
his  observations  should  be  taken  with  as  much  care  as  though 
he  were  ignorant;  and  that  great. precautions  must  be  taken  to 
avoid  the  almost  unconscious  tendency,  to  which  all  observers 
are  more  or  less  subject,  of  making  the  observations  correspond 
with  what  is  thought  to  be  the  truth. 

In  many  cases  artificial  devices  can  be  used  to  insure  unprejudiced 
observations.  For  example,  the  scale  of  a  micrometer  screw  may  be 
covered,  so  that  it  is  kept  out  of  sight  until  the  setting  is  made.  Or, 
in  an  experiment  like  that  on  the  Coefficient  of  Friction  (No.  d),  one 
experimenter  may  adjust  the  weights  while  the  other  observes  whether 
the  motion  obtained  is  uniform.  Since  the  latter  does  not  see  the 
weights,  his  judgment  is  uninfluenced  by  any  assumption  as  to  the  law 
by  which  they  vary. 

In  the  measurement  of  almost  all  physical  quantities  the 
results  will  be  better  if  the  observation  is  repeated  several 
times.  The  individual  observations  will  doubtless  differ  from 
one  another  on  account  of  slight  unavoidable  errors ;  but  the 
mean  of  the  results  will  in  all  probability  be  nearer  the  truth 
than  any  single  observation.  To  gain  the  advantages  of  taking 


4  JUNIOR   COURSE   IN   GENERAL  PHYSICS. 

an  average,  however,  it  is  necessary  that  each  observation 
should  be  independent  of  all  the  rest.  Knowing  that  all  the 
measurements  should  be  alike  except  for  accidental  errors, 
there  is  an  unconscious  tendency  to  make  them  agree.  This 
tendency  must  be  carefully  guarded  against,  as  in  the  cases 
cited  above.  Each  observation  should  be  taken  as  carefully 
as  though  the  final  result  depended  upon  it  alone. 

Estimation  of  Tenths.  —  In  measurements  in  which  a  grad- 
uated scale  of  any  kind  is  used  it  often  happens  that  the  result 
sought  cannot  be  expressed  by  any  exact  number  of  scale  divis- 
ions. For  example,  in  using  a  thermometer  graduated  to  single 
degrees,  the  top  of  the  mercury  column  will  probably  come 
between  two  divisions  on  the  scale.  In  such  cases  always 
estimate  the  fractional  part  of  a  division  by  the  eye,  expressing 
the  fraction  in  tenths.  Even  if  the  estimation  is  poor,  it  gives 
results  nearer  to  the  truth  than  if  the  fraction  were  disregarded ; 
while  after  a  little  practice  it  will  be  found  possible  to  estimate 
tenths  with  great  accuracy. 

Choice  of  Conditions.  —  It  often  happens  that  the  accuracy  of 
the  results  of  an  experiment  can  be  improved  by  a  proper 
choice  of  the  conditions  under  which  the  observations  are  made. 
An  example  of  this  fact  occurs  in  the  experiment  where  the 
internal  resistance  of  a  cell  is  determined  by  measurements  of 
the  current  sent  by  the  cell  through  two  different  external 
resistances.  If  Iv  Rv  and  72,  R^  represent  the  corresponding 
values  of  current  and  resistance,  the  internal  resistance  of  the 
cell  is 


It  is  evident  that  if  71  and  73  are  nearly  alike,  a  slight  error  in 
the  measurement  of  either  may  cause  a  very  large  error  in  x. 
To  make  the  results  reliable  it  is  therefore  necessary  to  choose 
R  and  R  so  that  the  two  values  of  the  current  shall  differ 


INTRODUCTION.  5 

widely.  There  are  many  cases  similar  to  this,  where  an  inspec- 
tion of  the  formula  by  which  the  results  are  to  be  computed 
will  suggest  what  conditions  will  make  the  influence  of  acci- 
dental errors  as  small  as  possible.* 

Computations. — In  computing  results  every  precaution  should 
be  used  to  avoid  simple  numerical  mistakes.  Mistakes  due  to 
careless  adding  or  subtracting,  to  incorrect  copying  from  one 
sheet  to  another,  to  the  misplacing  of  a  decimal  point,  etc.,  are 
a  source  of  great  annoyance,  and  unless  care  is  used  to  avoid 
them  they  will  appear  with  a  frequency  that  is  startling  to  one 
unaccustomed  to  computing.  The  best  safeguard  against  mis- 
takes is  neatness  and  an  orderly  arrangement  of  the  work.  In 
many  cases  four  or  five  place  logarithms  are  a  help,  not  so 
much  on  account  of  any  saving  of  time,  as  because  of  the  dimin- 
ished liability  of  mistakes.  Tables  of  squares,  reciprocals,  etc., 
can  often  be  used  to  advantage,  and  the  slide  rule,  when  one  is 
accustomed  to  its  use,  affords  a  considerable  saving  in  time  and 
worry.  When  a  number  of  similar  computations  are  to  be 
made,  the  work  should  be  done  systematically  and  the  results 
arranged  in  tabular  form. 

In  working  up  the  results  of  an  experiment  time  is  often 
wasted  by  carrying  the  results  to  a  degree  of  refinement  that 
is  not  warranted  by  the  observations  upon  which  the  computa- 
tions are  based.  Very  few  of  the  experiments  that  are  described 
here  will  give  results  that  are  accurate  to  within  less  than  one- 
tenth  of  one  per  cent.  In  most  cases,  therefore,  it  is  useless  to 
express  the  result  by  more  than  three,  or  at  most  four,  sig- 
nificant figures.  If  it  is  decided  from  an  inspection  of  the 
observations  that  the  result  should  be  carried  to  three  places, 
then  the  computations  should  be  made  with  four  places  in  order 
to  insure  the  accuracy  of  the  last  significant  figure  of  the  result. 


*  In  this  connection  see  the  paragraph  dealing  with  the  "  Influence  of  errors 
of  observation  upon  derived  results,"  p.  18. 


6  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

In  the  progress  of  the  work  numbers  may  be  obtained  in  which 
five  or  six  significant  figures  appear  ;  in  such  cases  all  beyond 
the  fourth  may  be  discarded. 

In  many  cases  approximate  methods  may  be  used  which  will 
effect  a  considerable  saving  in  time  without  diminishing  the 
accuracy  of  the  results.  For  example,  it  often  happens  that  a 

factor  of  the  form  -  appears  as  a  multiplier,  k  being  a  very 
\+k 

small  quantity.  In  most  cases  it  is  sufficiently  accurate  to  say 
that 

1 


and  in  general 

(i  +/£)"=  i  -\-nk  when  k  is  small.  * 

Units.  —  In  almost  all  physical  measurements,  the  units 
employed  are  based  upon  the  centimeter-gram-second  system. 
Since  this  system  differs  in  several  important  particulars  from 
that  generally  used  in  engineering  work,  it  is  essential  that  these 
differences  should  be  clearly  understood. 

In  physics,  all  derived  units  are  defined  in  terms  of  the  fun- 
damental units  of  length,  mass,  and  time.  In  the  foot-pound- 
second  system,  commonly  employed  in  engineering  work,  the 
fundamental  units  are  length,  weight,  and  time.  Now  the  terms 
"  weight  "  and  "  mass,"  although  technically  quite  different  in 
meaning,  are  frequently  confused  in  ordinary  conversation,  and 
it  is  probably  from  this  cause  that  the  relation  between  the 
two  systems  is  so  often  misunderstood. 

It  must  be  remembered  that  the  weight  of  a  body  is  defined 
as  the  force  with  which  the  body  is  pulled  downward  by  gravity. 
By  the  word  pound  is  meant,  not  the  block  of  metal  which  weighs 
a  pound,  but  the  force  by  which  that  block  is  drawn  toward  the 
center  of  the  earth.  Since  a  force  is  numerically  equal  to  the 
product  of  the  mass  moved  into  the  acceleration,  we  have 

*  Other  examples  of  the  use  of  approximations  will  be  found  in  Kohlrausch, 
Glazebrook  and  Shaw,  and  in  Stewart  and  Gee,  Appendix  to  vol.  i. 


INTRODUCTION. 


W  =  Mg,  and  in  order  to  find  the  mass  of  a  body  whose  weight 
in  pounds  is  known,  we  must  divide  the  weight  by  g\  i.e. 


The  mass  of  a  pound  weight  is  therefore  y^,  an<^  tne 

mass  in  the  foot-pound-second  system  is  the  mass  of  a  body 

which  weighs  32.2  Ibs. 

In  the  C.  G.  S.  system  the  gram  is  the  unit  of  mass.  By  the 
word  gram,  therefore,  is  meant  the  amount  of  matter  contained 
in  a  certain  standard  piece  of  metal.  The  weight  of  this  piece 
of  metal  is  found  by  multiplying  its  mass  by  the  acceleration  of 
gravity,  and  for  the  latitude  of  Ithaca  (about  40°)  is  a  little 
more  than  980  dynes. 

The  process  of  weighing  a  body  by  means  of  a  balance  con- 
sists of  choosing  the  weights  so  that  both  scale  pans  are  pulled 
downward  by  gravity  with  the  same  force.  When  the  adjust- 
ment is  correct,  the  weight  is  therefore  the  same  on  each  pan. 
But  since,  so  long  as  g  remains  unaltered,  the  mass  of  a  body  is 
proportional  to  its  weight,  the  two  masses  must  also  be  equal. 
The  balance  may  therefore  be  used  either  for  comparing 
weights,  or  for  comparing  masses.  In  physical  experiments 
the  weight  is  seldom  required,  so  that  the  balance  is  used 
almost  entirely  for  the  measurement  of  mass.  The  standards 
used,  being  grams,  or  multiples  of  a  gram,  are  standards  of 
mass,  and  the  term  "  weights,"  which  is  so  commonly  applied 
to  them,  is  really  a  misnomer. 

If  it  is  found,  therefore,  in  making  a  weighing  by  the  balance  that 
100  grams  are  required  to  produce  equilibrium,  the  mass  of  the  body 
weighed  is  shown  to  be  100  grams.-  The  weight  'of  the  body  is  100  x  g  = 
98,000  dynes. 

If  care  is  used  in  distinguishing  between  the  terms  "  weight  " 
and  "mass,"  no  difficulty  shouk}  be  experienced  in  passing  from 
one  system  of  units  to  the  other.  The  two  systems  are  per- 
fectly consistent  with  each  other  when  properly  used,  and  each 


8  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

has  special  advantages  for  the  kind  of  work  in  which  it  is  com- 
monly employed. 

Graphical  Representation  of  Results.  —  When  a  series  of 
observations  has  been  taken  to  show  the  manner  in  which  one 
quantity  depends  upon  another,  it  is  often  of  advantage  to 
present  a  summary  of  the  results  to  the  eye  by  means  of  a 
curve.  Points  upon  such  a  curve  are  located  on  cross-section 
paper  by  using  the  values  of  one  quantity  as  abscissas,  and  the 
corresponding  values  of  the  other  quantity  as  ordinates,  the 
scales  used  in  measuring  the  various  co-ordinates  being  any  that 
are  convenient.  It  is  customary  to  use  the  values  of  the  inde- 
pendent variable  as  abscissas. 

As  an  example  of  the  use  of  the  graphical  method,  we  may  consider 
the  experiment  on  the  coefficient  of  Friction  (Ci).  In  this  experi- 


5  K  1O  K  15  K  2O  K  25  K 

PRESSURE 

Fig.  1. 

ment  the  force  necessary  to  overcome  the  friction  between  iron  and  wood 
is  measured  for  a  number  of  different  values  of  the  pressure  between  the 
two.  It  is  natural  to  suppose  that  the  amount  of  friction  depends  in 
some  way  upon  the  pressure.  To  determine  the  law  of  this  dependence, 
a  curve  is  platted,  in  which  pressures  are  used  as  abscissas,  and  the 


INTRODUCTION.  9 

corresponding  values  of  the  friction  as  ordinates.  If  the  observations 
have  been  carefully  taken,  the  points  located  in  this  way  will  be  found 
to  lie  very  nearly  upon  a  straight  line  passing  through  the  origin.  If 
the  divergence  from  a  straight  line  is  not  great,  it  is  proper  to  assume 
that  such  divergence  in  the  case  of  individual  points  is  due  to  the 
accidental  errors  of  observation,  and  that  a  straight  line,  passing  as 
nearly  as  possible  through  all  the  points,  really  represents  the  relation 
sought.  Now  the  equation  of  a  straight  line  passing  through  the  origin 
is  y  =  mx,  in  which  m  is  a  constant.  But  the  x's  of  our  line  represent 
pressures,  while  the  ys  represent  the  corresponding  values  of  the 
friction.  The  law  established  by  the  experiment  is  therefore  that 
F=  mP;  i.e.  friction  is  proportional  to  pressure. 

It  is  to  be  observed  that  when  a  curve  is  platted  in  order 
to  show  the  relation  between  two  variables,  it  is  by  no  means 
necessary  that  the  horizontal  and  the  vertical  scale  should  be 
the  same.  Either  scale  may  be  assumed  at  pleasure,  and  with- 
out reference  to  the  other. 

In  the  case  just  cited,  for  example,  the  horizontal  scale  may  be 
taken  as  5  kilograms  to  the  inch,  while  the  vertical  scale  may  be  i 
kilogram,  -fa  kilogram,  or  any  other  quantity  that  proves  convenient. 
In  taking  readings  from  the  curve,  however,  regard  must  be  paid  to 
the  scale  employed.  If,  for  example,  the  horizontal  scale  adopted  is  5 
kilograms  to  the  inch,  5  inches  would  be  read  25  kilograms.  If  the 
vertical  scale  at  the  same  time  is  -fa  kilogram  to  the  inch,  5  inches  on 
the  vertical  scale  would  be  read  i  kilogram. 

The  example  referred  to  above,  where  the  curve  obtained  is 
a  straight  line  passing  through  the  origin,  illustrates  the  sim- 
plest case  that  could  arise.  In  other  cases  where  the  graphical 
method  is  used  the  curve  obtained  may  prove  to  be  a  straight 
line  which  does  not  pass  through  the  origin ;  or  it  may  be  any 
form  of  curved  line,  such  as  a  parabola  or  a  hyperbola.  In 
any  case  the  law  sought  is  determined  as  before,  and  can  be 
expressed,  either  in  words  or  by  a  formula,  as  soon  as  the  curve 
is  recognized. 

Since  the  straight  line  is  the  curve  which  is  most  readily 
tested,  it  is  often  convenient  to  transform  the  results  of  an 


10  JUNIOR   COURSE   IN    GENERAL   PHYSICS. 

experiment  in  such  a  way  that  they  will  give  a  straight  line 
when  platted. 

Suppose,  for  example,  that  the  volume  of  a  gas  has  been  measured 
when  subjected  to  a  number  of  different  pressures.  We  know  from 
Boyle's  Law  that  PV=  a  constant  =  k.  If  the  results  were  platted, 
therefore,  with  pressures  and  corresponding  volumes  for  co-ordinates, 
the  resulting  curve  would  be  a  hyperbola  whose  equation  is  xy  =  k.  If, 
however,  we  plat  instead  of  volumes  the  products  PV,  the  curve  will  be 
a  straight  line  with  the  equation  y  =  k.  By  observing  whether  this  line 
is  accurately  straight,  the  law  can  be  tested  more  readily  than  if  the 
first  curve  had  been  used,  while  if  the  line  is  not  straight  it  affords  a 
simple  means  of  exhibiting  the  deviation  from  Boyle's  Law  to  the  eye. 

If  the  method  described  in  Exp.  Hx  for  verifying  Boyle's  Law  is 
employed,  the  data  may  be  platted  in  still  a  different  way  to  advantage. 
In  this  method  the  total  volume  V  is  not  measured,  but  merely  a 
portion  v,  while  a  part  VQ  of  the  volume  remains  unknown,  but  constant. 


Then  F=(z;  +  z;0),  (i) 

P(v  +  v*)  =  k.  (2) 


If  now  P  and  V  are  taken  as  co-ordinates,  a  hyperbola  should  be 

obtained.     But  if  v  and  -  are  used,  the  resulting  line  should  be  straight, 

P 

its  equation  being 

ky  =  x  +  v<>.  (3) 

If  the  data  are  platted  in  this  way,  a  means  is  therefore  afforded  of 
determining  both  v0  and  k.  Since  the  line  obtained  is  straight,  we 
know  that  the  form  of  its  equation  must  be 

y  =  ax  +  b,  (4) 

and  the  numerical  values  of  a  and  b  can  be  at  once  computed.     From 
Boyle's  Law,  however, 


,  =  ^  +  f-  (5) 

Since  these  two  equations  represent  the  same  line,  we  must  have 

i  =  *,and=£.  (6) 


INTRODUCTION.  TI 

Graphical  methods  are  of  such  great  value  in  all  branches 
of  physical  investigation  that  their  use  is  recommended  in  a 
large  number  of  the  experiments  which  follow.  The  student  is 
strongly  advised  .to  make  himself  familiar  with  graphical  methods 
and  their  interpretation,  as  early  as  possible. 

Reports.  —  As  soon  as  the  observations  required  in  an  experi- 
ment have  been  completed,  and  the  results  computed,  a  report 
is  to  be  written,  describing  in  detail  the  work  that  has  been  done. 
This  report  should  be  sufficiently  clear  and  complete  to  enable 
it  to  be  understood  by  any  person  having  a  good  general  knowl- 
edge of  physics,  even  though  the  particular  experiment  described 
is  entirely  unfamiliar  to  him.  Each  report  should  therefore 
contain  the  following  : 

(1)  A  statement  of   the  object  of   the  experiment  and  an 
explanation  of  the  means  employed  to  accomplish  this  object. 

(2)  A  description  of  the  apparatus  used.* 

(3)  All  formulas  used,  which  express  relations  between  phys- 
ical quantities,  should  be  proven.f     The  object  of  putting  such 
demonstrations  in  the  report  is  to  make  it  clear  to  the  instructor 
that  the  principles  involved  are  fully  understood.     The  student 
will  find,  also,  that  there  is  no  better  way  of  making  a  subject 
perfectly  clear  to  himself  than  by  presenting  it  in  such  a  form 
as  to  be  readily  intelligible  to  some  one  else.     Those  steps  or 
details  of  a  demonstration  which  are  merely  referred  to  in  the 
text-books  should  therefore  be  very  clearly  explained.     Origi- 
nality in  the  methods  of  proof  is  desirable,  but  of  course  cannot 
be  expected  in  every  case. 

(4)  The  report  should  contain  all  the  original  data,  and  an 
indication  of  the  numerical  work  by  which  the  results  are  ob- 
tained.    It  is  not  necessary  to  include  all  the  computations  in 


*  In  case  the  same  apparatus  has  been  employed  in  previous  experiments,  how- 
ever, it  is  not  necessary  to  describe  it  a  second  time. 

t  The  proof  of  purely  mathematical  formulas,  such  as  the  trigonometrical  rela- 
tions used  in  solving  triangles,  is  not  required. 


12  JUNIOR   COURSE    IN   GENERAL   PHYSICS. 

the  report,  although  where  this  can  be  done  systematically  and 
neatly,  it  is  an  advantage.  In  case  the  results  are  obtained  by 
substitution  in  a  formula,  the  numerical  work  should  be  given 
in  detail  in  at  least  one  case. 

(5)  When  possible  the  results  obtained  should  be  compared 
with  the  results  of  previous  experiments  as  found  in  various 
reference  books. 

When  graphical  methods  have  been  used  in  connection  with 
an  experiment,  the  curves  obtained  are  to  be  included  in  the 
report.  In  such  cases  the  scale  by  which  the  co-ordinates  have 
been  measured  should  be  clearly  indicated  on  the  drawing  itself. 

In  writing  reports,  it  is  always  to  be  borne  in  mind  that  one 
important  benefit  which  practice  in  this  work  may  accomplish 
is  the  acquirement  of  clearness  and  facility  of  expression  in  the 
description  of  scientific  investigations.  The  arrangement  and 
wording  of  each  report  should  therefore  be  carefully  considered 
with  this  object  in  view. 


INTRODUCTION. 


ERRORS    OF    OBSERVATION    AND    METHOD    OF    LEAST    SQUARES. 

Sources  of  Error  in  Physical  Measurements. — All  physical 
measurements  are  subject  to  error  from  a  variety  of  sources. 
Although  the  choice  of  proper  methods,  the  employment  of 
carefully  constructed  instruments,  and  great  care  in  the  obser- 
vations themselves  may  enable  results  to  be  reached  which  are 
quite  close  to  the  truth,  yet  absolute  accuracy  can  in  no  case 
be  expected.  The  effort  of  the  experimenter  should  always 
be  to  reduce  these  errors  to  a  minimum ;  yet  he  may  feel  per- 
fectly sure  that  to  completely  eliminate  them  is  quite  impossible. 

As  an  example  of  the  different  ways  in  which  inaccuracies  can  occur, 
we  may  consider  a  case  which  represents  probably  the  simplest  measure- 
ment imaginable ;  namely,  the  measurement  of  a  length  by  means  of  a 
graduated  scale.  The  chief  sources  of  error  in  this  measurement  may 
be  summarized  as  follows  : 

1.  The  scale  may  be  incorrect  either  in  total  length  or  in  graduation. 

2.  Even  if  it  were  possible  that  the  scale  were  constructed  with 
perfect  accuracy,  it  can  only  be  correct  at  one  definite  temperature. 
The  coefficient  of  expansion  of  the  scale   must  therefore  be  known, 
while  its  temperature  must  be  determined  at  the  instant  of  making  the 
measurement.     Two  sources  of  error  are  here  introduced. 

3.  The  end  of  the  length  to  be  measured  will  in  all  probability  lie 
between  two  divisions  of  the  scale.     The  fractional  part  of  a  scale  divis- 
ion must  therefore  be  estimated,  and  on  account  of  a  variable  illumina- 
tion of  the  scale,  an  improper  location  of  the  observer's  eye,  or  lack  of 
experience  on  the  part  of  the  experimenter,  this  estimation  is  always 
subject  to  error. 

4.  Lastly,  the  observer  may  make  a  mistake ;  i.e.  may  read  10  for 
20,  Jg-  for  ^,  etc. 

A  little  consideration  will  show  that  all  possible  errors  may 
be  made  to  fall  under  four  classes  : 

1.  Errors  of  method. 

2.  Inaccuracies  in  instruments. 

3.  Accidental  errors  of  observation. 

4.  Mistakes. 


I4  JUNIOR   COURSE   IN   GENERAL  PHYSICS. 

The  avoidance  of  errors  due  to  the  employment  of  faulty 
methods  is  largely  a  matter  of  judgment  and  experience  on 
the  part  of  the  experimenter.  No  general  rule  can  be  given. 
Probably  the  best  means  of  testing  for  the  presence  of  errors 
in  the  method  of  measurement  employed  is  to  repeat  the  deter- 
mination by  several  radically  different  methods.  If  the  results 
agree,  it  is  to  be  presumed  that  the  methods  contain  no  funda- 
mental errors. 

The  presence  of  inaccuracies  in  the  instruments  used  may 
similarly  be  tested  by  making  the  same  measurement  with 
several  different  instruments.  Special  methods  may  also  in 
most  cases  be  devised  by  which  the  errors  of  any  given  instru- 
ment may  be  determined.  These  methods  are  different  for 
each  particular  case,  so  that  it  is  useless  to  give  illustrations 
here. 

After  the  errors  of  method  and  of  apparatus  are  as  far  as 
possible  eliminated,  there  still  remain  the  "  accidental  errors 
of  observation."  Two  measurements  of  the  same  quantity 
made  by  the  same  observer,  with  the  same  instrument,  and  to 
all  appearances  under  the  same  conditions,  will,  in  the  great 
majority  of  cases,  differ  from  each  other  by  an  appreciable 
amount.  Such  discrepancies  are  entirely  accidental,  and  a 
cause  for  the  disagreement  in  the  two  results  can  in  no  case 
be  assigned.  The  discussion  of  these  errors  can  therefore  only 
be  undertaken  with  the  aid  of  the  theory  of  probabilities,  and 
numerous  treatises  have  in  fact  been  written  which  deal  with 
the  "theory  of  errors"  and  the  "  method  of  least  squares."* 
The  principal  results  of  such  discussion,  in  so  far  as  they  have 
an  application  to  physical  measurements,  will  be  briefly  stated 
here. 


*  Merriman,  Method  of  Least  Squares;  Violle,  Cours  de  Physique  (see  Introduc- 
tion to  vol.  i)  ;  Weinstein,  Handbuch  der  Physikalischen  Maassbestimmungen,  vol.  I ; 
Holman's  Precision  of  Physical  Measurements;  also,  for  brief  discussions,  Kohlrausch, 
Physical  Measurements,  and  the  Appendix  to  Stewart  and  Gee,  vol.  I. 


INTRODUCTION.  15 

Probable  Error,  etc.  —  If  a  large  number  of  independent* 
measurements  of  the  same  quantity  are  made,  it  is  evident  that 
one  result  is  as  likely  to  be  correct  as  any  other.  As  a  matter 
of  fact,  all  of  the  results  are  doubtless  in  error.  It  is  also 
evident  that  the  most  probable  value  of  the  result  sought  will 
be  found  by  taking  the  average  of  all  the  values  found.  This 
average  will  probably  be  more  correct  than  any  one  of  the 
single  determinations.  For  this  reason  it  is  always  advisable 
to  repeat  a  determination  a  number  of  times  when  the  condi- 
tions are  such  as  to  make  this  possible. 

If  a  series  of  independent  observations  has  been  taken  under 
favorable  conditions  and  by  a  skillful  observer,  so  that  the  indi- 
vidual results  do  not  differ  greatly  from  one  another,  it  is  obvious 
that  the  average  has  greater  probable  accuracy  than  if  the  con- 
ditions had  been  unfavorable  so  that  the  individual  results 
showed  a  wide  divergence  among  themselves.  From  an  inspec- 
tion of  a  series  of  determinations  we  may  therefore  form  an  esti- 
mate of  the  probable  accuracy  of  the  average.  In  order  to  express 
this  estimate  numerically  the  term  "  Probable  Error  "  has  been 
introduced,  which  is  defined  as  follows  : 

The  probable  error  of  a  result  is  a  quantity  e  such  that  the 
probability  that  the  actual  error  is  greater  than  e  is  the  same  as 
the  probability  that  the  actual  error  is  less  than  e.  f 

A  result  whose  probable  error  is  small  is  thus  in  all  proba- 
bility more  accurate  than  one  whose  probable  error  is  large. 

The  probable  reliability  of  a  result  is  often  indicated  by  writing  the 
probable  error  with  the  sign  ±  after  the  result  itself:  e.g.  7=27.36  ±0.21. 


*  Too  much  stress  cannot  be  laid  on  the  condition  that  the  observations  must  be 
independent ;  i.e.  the  observer  must  be  entirely  uninfluenced  by  results  previously 
obtained,  or  by  his  own  opinion  as  to  what  the  result  "  ought "  to  be.  The  avoidance 
of  this  bias  in  making  a  series  of  readings  of  the  same  quantity  is  one  of  the  most 
difficult  things  which  an  observer  has  to  learn. 

f  The  name  "  probable  error  "  is  an  unfortunate  one  and  is  apt  to  lead  to  confu- 
sion. That  the  probable  error  of  a  result  is  e  does  not  mean  that  the  result  is  proba- 
bly in  error  by  this  amount. 


i6 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


If  another  series  of  measurements  of  the  same  quantity  gave  the  result 
/=  27.51  ±0.38,  it  is  clear  that  the  first  result  is  more  reliable. 

If  a  series  of  observations  av  a2,  ...,  an  has  been  taken,  the 
average  being  a,  then  the  probable  error  of  the  average  may  be 
shown  to  be  * 


e  = 


»(«-!) 

It  may  happen  that  it  is  desired  to  determine  the  probable 
accuracy  of  the  result  obtained  from  a  single  reading.  The 
probable  error  of  a  single  observation  is  given  by  the  formula 


=  ±  o.67449\/(a 

w 


n  —  I 


(8) 


That  is  to  say,  if  a  single  observation  of  the  quantity  in 
question  is  made,  the  error  is  as  likely  to  be  greater  than  e1  as 
it  is  to  be  less. 

As  an  example  of  the  computation  of  the  probable  error  we  may 
consider  the  following  case  where  ten  independent  settings  are  made 
with  a  spherometer  on  the  same  surface.  (See  Exp.  Aj.) 


Readings  of  micrometer. 

Deviation  (d)  from  the  mean. 

* 

3.445  mm. 

—  0.001 

0.00000  1 

3.448 

+  0.002 

0.000004 

3-442 

—  O.OO4 

0.000016 

3-45° 

+  0.004 

0.000016 

3-45  * 

+  0.005 

0.000025 

3-444 

—  0.002 

0.000004 

3.446 

±  o.ooo 

o.oooooo 

3-442 

—  0.004 

0.000016 

3-445 

—  0.00  1 

O.OOOOOI 

3-447 

+  o.ob  i 

0.00000  1 

Mean  3.446 

2d2  =  0.000094 

Probable  error  of  the  mean  e  =  .0007 

Probable  error  of  single  observation  e'  —  .0022 

*  For  the  derivation  of  this  formula,  see  any  text-book  of  least  squares. 

It  is  to  be  observed  that  the  computation  of  the  probable  error  has  no  signifi- 
cance unless  n  is  large.  Unless  at  least  ten  observations  have  been  taken,  it  is 
useless  to  compute  e. 


INTRODUCTION.  !7 

On  account  of  the  annoyance  in  computing  the  probable 
error,  the  "average  deviation"  is  often  used  instead;  i.e.  the 
average  (disregarding  signs)  of  the  deviations  of  the  individual 
observations  from  the  mean. 

It  is  to  be  observed  that  the  probable  error  affords  no  means 
of  estimating  the  so-called  "  constant  errors  "  that  are  caused 
by  improper  methods  of  measurement  or  by  imperfections  in 
the  instruments  used.  These  may  be  very  large  even  when  the 
"probable  error"  is  quite  small.  The  use  of  the  "probable 
error"  may  be  looked  upon  as  merely  an  arbitrary  means  of 
showing  at  a  glance  how  closely  the  individual  observations 
have  agreed  among  themselves,  and  it  indicates,  therefore,  to 
what  extent  the  accidental  errors  of  observation  have  been 
eliminated. 

Assignment  of  Weights  in  taking  an  Average.  —  When  the 
same  quantity  has  been  measured  by  several  different  methods, 
the  results  will  in  general  differ,  and  it  is  often  desirable  to 
combine  all  the  results  by  taking  an  average.  In  such  cases 
"weights"  should  be  assigned  to  the  different  determinations 
in  accordance  with  their  probable  accuracy.  The  theory  of 
probabilities  shows  that  in  taking  an  average,  each  quantity 
should  be  given  a  weight  equal  to  the  reciprocal  of  the  square 
of  its  probable  error ;  i.e.  if  the  various  values  determined  by 
the  different  methods  are  Av  A2,  Az,  etc.,  the  probable  errors 
being,  respectively,  elt  e2,  eB,  etc.,  the  most  probable  value  of  the 
quantity  in  question,  as  determined  from  all  of  the  observa- 
tions, is 


(9) 


In  Exp.  Aj,  for  example,  the  length  /  of  one  side  of  the  triangle 
formed  by  the  three  legs  of  the  spherometer  may  be  determined  in 
several  different  ways.  Let  the  result  obtained  by  one  method  be 
/=  6.1  2cm  ±  0.03,  while  that  determined  by  another  and  less  accurate 


IS  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

method  is  /=  6.2Ocm  ±  o.i  i.    It  is  certainly  not  right  to  use  the  average 
.12  +    •20(==6tI6),  for  much  more  reliance  can  be  placed  on  the 

2 

first  result  than  on  the  second.     According  to  the  rule  above  stated  the 
most  probable  value  of  /  is 


1        i 


(o.03)2      (o.n)2 

Influence  of  the  Errors  of  Observation  upon  Derived  Results.  — 

It  often  happens  that  the  final  result  sought  must  be  computed 
from  the  observations  themselves  by  substitution  in  some 
formula.  In  such  cases  it  is  of  importance  to  know  how  the 
final  result  will  be  influenced  by  possible  errors  in  the  indi- 
vidual observations.  If  an  error  in  one  of  the  quantities 
involved  will  produce  a  large  error  in  the  result,  then  this 
quantity  must  be  observed  with  especial  care.  On  the  other 
hand,  if  an  error  in  another  of  the  observed  quantities  has  only 
a  slight  influence  on  the  result,  it  is  needless  to  occupy  one's 
time  in  measuring  this  quantity  with  a  high  degree  of  refine- 
ment. By  considering  this  question  before  the  actual  meas- 
urements are  begun,  it  is  thus  possible  not  only  to  obtain 
better  final  results,  but  also  to  save  time  in  the  observations 
themselves. 

The  general  case  may  be  discussed  as  follows :  Let  the 
result  R,  which  is  sought,  be  some  function  <£  of  the  quantities 
to  be  observed ; 

i.e.  R  =  <f>(x,y,  2,  •••). 

Now  if  x,  y,  2,  etc.,  are  measured  with  absolute  accuracy,  R 
will  be  correct.  But  if  one  of  the  quantities  x  is  in  error  by 
the  amount  e,  then  an  error  Ex  will  be  introduced  into  the 
result,  and 

Ex  =  <j>(x+elty,  z,  •••)  —  $  (x,y,  z,  •••).  (11) 


INTRODUCTION.  !9 

>ince  e  will  in  general  be  quite  small  in  comparison  with  x, 
no  great  inaccuracy  will  be  introduced  by  treating  it  as  an 
infinitesimal ;  i.e.  neglecting  powers  higher  than  the  first : 

-T-t  7-  d       ,    /  \ 

Then  Ex  =  e1— -<b(x,  y,  z,  •••). 

ax 

Similarly,  Ey  =  e2—(j)  (x,  y,  s,  •••),  (12) 

E  =e  — 
z     e*dz 

etc. 

If  the  probable  errors  ev  e^  etc.,  are  known,  the  correspond- 
ing errors  Ext  Ey,  etc.,  in  the  result  may  thus  be  readily  com- 
puted. The  probable  error  in  the  result  due  to  the  combined 
effect  of  the  errors  in  all  of  the  observed  quantities  may  then 
be  shown  to  be  * 

E=vEx  +Ey  H      .  (13) 

Take,  for  example,  the  case  which  occurs  in  Exp.  Ax.  The  radius  of 
curvature  is  given  by  the  formula 


Let  us  suppose  that  /=  7.14  ±  0.05,  and  a  =  0.423  ±  0.004.     Sub- 
stituting in  the  formula  /=  7.14  and  a  =  0.423,  we  obtain 

r=  20.09. 
To  compute  Ea  and  £lt  we  have  : 

*.  =  ,'*(«,  /)  =  ,.£[£+«]  (15) 

da  da    o  #      2 


x0.423 


*  See  Merriman's  Method  of  Least  Squares,  etc. 


20  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


=  6l—  =  0.05  x      7'14  —  =  0.28. 
3#  3x0.423 

(17) 
/.  r=  20.09  ±  0.34. 

When  the  formula  is  known  by  means  of  which  the  result  of 
an  experiment  is  to  be  computed,  it  is  often  possible  to  deter- 
mine the  most  favorable  conditions  before  beginning  the  obser- 
vations. The  method  of  procedure  in  such  cases  can  best  be 
explained  by  means  of  the  following  example  :  * 

A  tangent  galvanometer  is  to  be  used  in  measuring  a  current.    What 
are  the  most  favorable  conditions  for  making  the  measurement  ? 
The  formula  for  computing  the  result  is  : 

/=/0tan<9.  (18) 

The  only  observed  quantity  is  0  ;  and  if  this  angle  is  read  from  a 
circular  scale,  it  is  liable  to  the  same  error,  e,  no  matter  from  what  part 
of  the  scale  it  may  be  read.  Let  the  resulting  error  in  /  be  E. 


Then  ^  =        ./otane=__.  (I9) 

The  relative  error  is 

£'=2=^  +  fotsmO  =  t-r-l  -  7=-r^4«  (20) 

/      cos2  0  sin  0  cos  0      sin  2  6 

It  is,  however,  evident  that  E1  reaches  its  smallest  value  when  sin  2  0 
reaches  its  greatest  value,  namely  unity.  In  other  words,  the  relative 
error  in  the  result  will  be  least  when  the  deflection  of  the  galvanometer 
is  45°.  If  the  galvanometer  used  has  several  coils,  these  should  there- 
fore always  be  so  connected  as  to  make  the  deflection  as  near  45°  as 
possible. 


*  Numerous  instructive  examples  will  be  found  discussed  in  detail  in  Holman's 
Discussions  of  the  Precision  of  Physical  Measurements. 


INTRODUCTION. 


21 


Determination  of  Constants  by  the  Method  of  Least  Squares. 

-  It  often  happens  that  a  series  of  observations  is  made,  not  of 
the  same  quantity,  but  of  quantities  which  are  known  to  be 
related  to  one  another.  If  the  form  of  the  equation  expressing 
this  relation  is  known  (as  is  usually  the  case),  the  question  then 
arises  as  to  what  values  should  be  given  to  the  constants  of  the 
equation  in  order  that  it  should  represent  the  results  of  experi- 
ment as  accurately  as  possible. 

A  case  of  this  kind  is  illustrated  by  Exp.  C2,  where  a  series  of 
observations  is  made  to  determine  the  relation  between  " power"  and 
"load  "  in  the  case  of  a  wheel  and  axle.  If  the  results  are  platted,  the 
points  corresponding  to  the  different  observations  will  probably  be  found 
to  lie  nearly  in  a  straight  line,  as  shown  in  Fig.  2.  Although  it  is  impos- 


Fig.  2. 

sible  to  draw  a  straight  line  which  shall  pass  through  all  the  observed 
points,  yet  it  seems  probable  that  these  points  would  have  formed  a 
straight  line,  had  it  not  been  for  accidental  errors  of  observation.  The 
problem,  therefore,  is  to  draw  a  straight  line  which  shall  pass  as  nearly 
as  possible  through  all  the  points.  This  can  often  be  done  by  the  eye ; 
but  when  the  highest  degree  of  accuracy  is  required,  the  Method  of 
Least  Squares  should  be  used  as  explained  below. 

The  method  of  procedure  in  all  such  cases  rests  upon  the 
principle*  that    the   results  will    best    be    represented    by  the 


*  For  the  proof  of  this  principle  see  any  text-book  of  Least  Squares. 


22  JUNIOR   COURSE    IN    GENERAL   PHYSICS. 

equation  in  question  when  the  constants  are  so  chosen  that  the 
sum  of  the  squares  of  the  deviations  of  the  individual  observa- 
tions from  the  values  computed  from  the  equation  is  a  minimum. 

In  the  example  just  cited  the  formula  which  expresses  the  relation 
between  "power"  (y)  and  "load"  (x)  is  evidently 

y  =  ax  +  b;  (21) 

for  the  observations  when  platted  have  been  found  to  give  roughly  a 
straight  line,  and  the  general  equation  of  a  straight  line  is  of  the  form 
stated.  As  the  result  of  experiment  a  number  of  values  ylf  y2,  ys,  etc., 
of  the  power  have  been  observed,  corresponding  respectively  to  loads  of 
x\j  x*,  #3,  etc.,  kilos.  Now,  if  the  constants  a  and  b  were  known,  it 
would  be  possible  to  compute  y  from  x  :  e.g. 

y\  =  axl  -f  b, 

yj  =  ax2  +  b,  (22) 

etc. 

[The  /s  have  been  primed  in  order  to  distinguish  them  from  the 
observed  values  ylt  y2,  etc.] 

The  principle  of  least  squares,  which  has  been  stated  above, 
now  says  that  a  and  b  must  have  such  values  that  the  sum 

to  -*')*+  to  -*')s  +  - 

shall  be  a  minimum. 

Interpreted  graphically,  this  means  that  the  line  must  be  so 
drawn  that  the  sum  of  the  squares  of  the  distances  i  Pv  2  Pv 
etc.,  shall  be  as  small  as  possible.  (See  Fig.  2.) 

To  determine  a  and  b  it  is  therefore  merely  necessary  to 
apply  the  ordinary  methods  for  maxima  and  minima  : 

(y\  -JV)2  +  (y*  -72')2  +  —  •»  2  (y  -y)2  =  a  minimum. 
But  y  =  ax  +  b. 


=  a  mnimum. 

it  is  to  be  observed  that  x^y-^  ^y^  etc.,  are  not  variables, 
but  constants,  being  the  quantities  determined  by  observation. 


INTRODUCTION.  23 

It  is  a  and  b  that  must  be  varied  until  such  values  are  found 
that  the  above  expression  is  a  minimum.* 
The  conditions  are  therefore  that 

—  2  (y  -  ax  -  bj  =  o  and  —  -  S  (y  -  ax  -  b?  =  o.        (23) 
da  do 

On  performing  the  differentiation  the  following  equations 
result  : 

-  2  (y\  —  ax\  —  b)*\  -  2  (y*  —  axi  —  fyx^  -  "- 

=  —  2  IX  (  y  —  ax  —  b]x  =  o, 

(24) 

-  2  (yl  -  ax^  -  b)  -  2  0/2  -  ax<i  -  b)  ---- 

=  —  2  5  (y  —  ax  —  b]  =  o. 

These  equations  may  be  more  readily  utilized  if  written  in 
the  following  form  :  f 


=  o, 

A  <25) 

—  on  =  o. 

In  the  last  equation  n  represents  the  number  of  observations. 

Since  the  quantities  ^xyy  2a?,  etc.,  are  readily  computed 
from  the  observations,  these  two  equations  make  possible  the 
determination  of  both  a  and  b.  In  fact, 


__ 

~      x-ri* 

(26) 


and  b  = 


*  Note  that  this  variation  of  a  and  b  in  the  algebraic  work  corresponds  to  shift- 
ing the  line  AB  in  the  graphical  consideration  of  the  problem.  In  the  one  case  a 
and  b  are  varied  until  certain  mathematical  considerations  indicate  that  ^(y—  X)2 
has  reached  a  minimum;  in  the  other  case  the  line  is  shifted  until  it  looks  to  the  eye 
as  though  a  good  intermediate  position  had  been  reached. 

t  The  student  is  cautioned  in  regard  to  the  use  of  the  sign  of  summation. 
*S,xy  means  xlyl  +  ^2_y2  +  •••,  while  2y  =  y\  +  y<i  +^3  -f  ••••  ~2>xy  is  therefore  not 
equal  to  2.r2y. 


24  JUNIOR   COURSE    IN   GENERAL   PHYSICS. 

In  the  general  case,  where  the  relation  between  x  and  y  is 
expressed  by  an  equation  of  any  form,  the  method  of  pro- 
cedure is  the  same  as  that  illustrated  by  the  example  above. 

Let  y  =  $(x,  a,  b,  c,  •••), 

where  a,  b,  c,  etc.,  are  the  constants  to  be  determined. 

The  observed  values  of  y  are  then  ylt  j/2,  ys,  etc.,  while  the 
computed  values  are  y±,  j2',  etc.  The  principle  of  least  squares 
requires  that  the  sum 

0/i~J/)2+(j2~~-7/2/)2H  —  shall  be  a  minimum  ; 
i.e.         [.7i-0(*i,  a,  b,  c,  —  )]2+|>2-<A(*2»  «,  ft  c,  ...)]2+... 

=  ^[y—  <$>(x,  a,  ft  c,  •••)]2=  a  minimum. 


*(  )]2=o,  (27) 

~2Lr-<M  )]2=o, 

etc. 

The  number  of  equations  obtained  in  this  way  is  always  equal 
to  the  number  of  constants  sought,  so  that  the  problem  is  in  all 
cases  determinate. 

In  applying  the  method  of  least  squares,  the  numerical  work 
is  always  somewhat  tedious,  especially  when  the  number  of 
observations  is  large.  For  this  reason  the  computations  should 
be  made  with  especial  care  ;  for  if  a  mistake  occurs,  considerable 
difficulty  will  be  met  with  in  discovering  it.  The  following 
example  illustrates  a  systematic  way  of  arranging  the  compu- 
tations, which  will  be  found  of  advantage  : 

Equation  is  known  to  be  of  the  form  y  =  ax-\-b. 


_ 
~ 


(  8) 


INTRODUCTION. 


X. 

y- 

xy. 

X-. 

5 

O.2O 

I.OO 

25 

IO 

0-34 

340 

100 

15 

0.48 

7.20 

225 

20 

0.64 

1  2.80 

400 

25 

0.80 

2O.OO 

625 

30 

0-93 

27.90 

900 

35 

I.IO 

38.50 

1225 

40 

1.24 

49.60 

1600 

45 

1.38 

62.IO 

2025 

5° 

i-54 

77.00 

2500 

275 

8.65 

299-5 

9625 

275 

8.65 
299.5 
9625 


^275^65-10x299.5^^ 
275   -10x9625 


^^275x299.5-8.65x9625^ 

10x9625 


0-433- 


The  equation  which  most  accurately  represents  the  relation 
between  the  quantities  measured  is  therefore 

7=0.299^+0.433. 

A  simple  means  of  detecting  large  mistakes  in  computation 
is  always  afforded  by  platting  the  curve  represented  by  the 
equation  found  by  least  squares  upon  the  same  diagram  as 
the  original  data.  This  curve  should  then  pass  close  to  all  of 
the  observed  points,  although  it  may  not  actually  pass  through 
any  one  of  them. 


CHAPTER   I. 

GROUP  A:  LENGTH,   TIME,  AND  MASS. 

(Aj)  Curvature  of  a  lens ;  (A2)  The  cathetometer ;  (A3)  Calibra- 
tion of  a  thermometer  tube ;  (A4)  Volumes  and  densities  by 
measurement ;  (A5)  Time  of  periodic  motion. 

EXPERIMENT  Ax.  Measurement  of  the  curvature  of  a  lens 
by  means  of  the  spherometer. 

The  spherometer,  as  indicated  by  its  name,  is  intended 
primarily  for  the  determination  of  the  radius  of  a  spherical  sur- 
face. It  can  also  be  employed,  however,  for  other  measurements : 


Fig.  3.  — The  Spherometer. 

for  example,  the  thickness  of  plates  of  glass  or  other  materials 
can  be  determined  by  means  of  the  spherometer,  although  unless 

26 


LENGTH,   TIME,  AND   MASS.  2/ 

the  plates  are  almost  perfectly  plane,  the  results  will  not  possess 
a  high  degree  of  accuracy. 

As  will  be  seen  by  reference  to  Fig.  3,  which  represents  a 
simple  type  of  spherometer,  the  instrument  consists  essentially 
of  four  metallic  rods  connected  in  the  manner  shown,  each  rod 
being  sharply  pointed  at  the  lower  end.  Three  of  these  rods 
are  fixed  in  position,  and  constitute  a  tripod  upon  which  the 
instrument  rests.  The  three  supporting  points  are  made  to  form 
an  equilateral  triangle.  The  fourth  rod  may  be  moved  in  a 
direction  at  right  angles  to  the  plane  of  this  triangle  by  means 
of  a  micrometer  screw. 

In  using  the  spherometer  to  determine  the  curvature  of  a 
surface  (see  Fig.  4)  which  is  known  to  be  spherical,  such  for 
example  as  that  of  a  lens,  the  reading  of  the  micrometer  is 


Fig.  4. 

taken,  first  when  the  instrument  rests  upon  a  plane  surface,  and 
then  when  it  is  placed  upon  the  lens  in  question.  All  four  points 
must  in  each  case  be  in  contact  with  the  surface.  The  difference 
between  the  two  micrometer  readings  then  gives  the  height  of 
the  fourth  point  above  the  plane  of  the  other  three.  If  this 


28  JUNIOR   COURSE   IN    GENERAL   PHYSICS. 

height  be  represented  by  a,  and  the  length  of  one  side  of  the 
triangle  formed  by  the  three  fixed  points  by  /  (l=pp'\  Fig.  4), 
then  the  radius  of  curvature  is  readily  shown  to  be  * 

r--2-44  (29) 

6a     2 

In  determining  /,  two  methods  may  be  employed ;  viz. 
(i)  the  three  sides  of  the  triangle  may  be  measured  directly  by  a 
millimeter  scale  ;  (2)  the  instrument  may  be  placed  upon  a  piece 
of  paper,  and  the  distances  between  the  impressions  left  by  the 
feet  may  be  measured.  After  making  a  number  of  measurements 
by  both  methods,  estimate  the  degree  of  accuracy  attainable  by 
each,  and  obtain  the  mean,  giving  to  each  measurement  its 
proper  weight.f 

Since  a,  which  enters  as  numerator  in  the  formula  for  r,  is 
in  general  a  very  small  quantity,  its  value  must  be  determined 
with  especial  care.  It  is  therefore  advisable  to  make  a  number 
of  independent  settings  of  the  micrometer,  and  to  use  the  mean 
of  the  readings  obtained.  Make  at  least  ten  readings  for  the 
plane  surface,  and  the  same  number  for  the  spherical  surface. 
Compute  in  each  case  the  " probable  error  of  the  mean"  and 
the  "  probable  error  of  a  single  observation." 

After  the  value  of  r  has  been  computed,  determine  the 
influence  upon  the  result  of  the  probable  error  in  measuring  the 
vertical  height ;  also  the  influence  of  the  error  which  is  liable  to 
occur  in  measuring  the  distance  between  the  legs  ;  and  finally 
the  probable  error  in  the  result,  arising  from  both  these  causes. 

In  making  a  series  of  readings  with  the  spherometer  upon 
the  same  surface,  it  is  best  not  to  look  at  the  scale  until  the 
setting  has  been  made.  Otherwise  it  is  difficult  to  avoid  being- 
influenced  by  a  knowledge  of  previous  readings.  Each  reading 


*  For  a  more  detailed  description  of  the  spherometer  and  derivation  of  this  for- 
mula, see  Stewart  and  Gee,  vol.  I. 
t  See  Introduction,  p.  1 7. 


LENGTH,   TIME,   AND   MASS.  2C> 

should  be  the  result  of   an  independent  attempt  to  obtain  an 
exact  setting. 

EXPERIMENT  A2.  Adjustment  of  a  cathetometer,  and  deter- 
mination of  the  sensitiveness  of  the  level. 

A  general  description  of  the  cathetometer  will  be  found  in 
any  text-book  of  physics.  The  various  adjustments  should  be 
made  in  the  following  order. 

I. 

To  make  the  level  parallel  to  the  axis  of  the  telescope.  — Adjust 
the  vertical  column  and  bring  the  bubble  to  the  middle  of  the 
level  tube  ;  then  reverse  the  telescope  in  its  Y's.  If  the  bubble 
settles  away  from  the  middle,  it  must  be  brought  back,  partly  by 
the  screws  that  attach  the  level  to  the  telescope,  and  partly  by 
changing  the  direction  of  the  telescope  itself.  Repeat  until  the 
bubble  remains  in  the  middle  of  the  tube  when  the  telescope 
is  reversed. 

II. 

To  adjust  the  telescope  to  a  right  angle  with  the  colttmn, 
and  to  make  the  column  vertical.  — Unclamp  the  column  and  turn 
it  till  the  telescope  is  parallel  to  the  line  joining  two  of  the 
leveling  screws  at  the  base ;  bring  the  bubble  to  the  middle  of 
the  tube,  and  then  turn  the  column  through  180°.  If  the  bub- 
ble moves  from  the  center,  it  must  be  brought  back,  partly  by 
the  screw  that  adjusts  the  angle  between  telescope  and  column, 
and  partly  by  the  leveling  screws  at  the  base,  using  only  the 
two  to  which  the  telescope  is  parallel.  Turn  now  through  90°, 
and  adjust  the  third  leveling  screw.  'Turn  back  to  the  first 
position,  and  repeat  the  adjustments  till  the  bubble  will  remain 
in  the  middle  of  the  tube  for  the  entire  revolution. 


III. 

To    adjust    the    line    of   collimation.  —  Bring  the  point  ; 
crossing  of   the   spider   lines    exactly   upon    some   well-define 


30  JUNIOR   COURSE    IN   GENERAL   PHYSICS.  . 

point  and  turn  the  telescope  upon  its  axis.  If  the  spider  lines 
move  away  from  the  point,  they  must  be  brought  back,  partly 
by  the  small  screws  in  the  ring  near  the  eye-piece,  and  partly 
by  moving  the  telescope.  Repeat  until  the  point  of  intersection 
of  the  spider  lines  remains  fixed  while  the  telescope  is  rotated. 

IV. 

To  determine  the  angular  value  of  one  division  of  the  level. — 
Incline  the  telescope  as  far  as  possible,  at  the  same  time  mak- 
ing sure  that  the  position  of  both  ends  of  the  bubble  can  be 
read  on  the  scale ;  raise  or  lower  the  telescope  till  the  inter- 
section of  the  spider  lines  coincides  with  some  well-defined 
point,  and  take  the  reading  on  the  vertical  scale ;  then  incline 
the  telescope  in  the  other  direction,  and  raise  or  lower  it  till  the 
spider  lines  again  coincide  with  the  fixed  point ;  read  the  ver- 
tical scale,  and  measure  the  distance  from  it  to  the  point.  This 
may  be  taken  as  the  radius  of  a  circle,  of  which  the  difference 
in  readings  upon  the  vertical  column  may  be  considered  as  an 
arc.  The  angle  subtended  by  it  may  then  be  computed.  This, 
divided  by  the  number  of  divisions  through  which  the  bubble 
has  moved,  is  the  angle  sought. 

EXPERIMENT  A3.     Calibration  of  a  thermometer  tube. 

The  object  of  this  experiment  is  to  determine  how  completely 
the  variations  in  the  bore  of  a  thermometer  tube  have  been 
corrected  by  the  graduation  of  its  scale.  The  experiment  is 
also  of  value  in  affording  practice  in  the  use  of  the  dividing 
engine. 

As  usually  employed  in  the  Physical  Laboratory,  the  divid- 
ing engine  is  merely  an  instrument  for  the  accurate  measure- 
ment of  lengths.  The  more  or  less  complicated  modifications 
which  make  it  possible  to  use  the  dividing  engine  in  construct- 
ing scales  and  in  ruling  diffraction  gratings  need  not  be  here 
considered. 

The  essential  parts  of  one  of  the  common  forms  of  dividing 


LENGTH,   TIME,   AND   MASS.  3! 

engine  are  shown  in  Fig.  5.  The  most  important  part  of  the 
instrument  is  the  carefully  constructed  screw,  which  extends 
the  whole  length  of  the  " engine."  The  figure  shows  only  a 
portion  of  the  screw  and  bed.  The  reading  microscope  m  is 
attached  to  a  carriage  which  rests  upon  a  nut  fitting  this  screw. 
By  the  rotation  of  the  latter,  the  carriage  can  be  moved  through 
any  distance  within  the  range  of  the  instrument.  If  the  pitch 
of  the  screw  is  known,  the  distance  through  which  the  micro- 
scope has  been  moved  can  be  computed  from  the  number  of 
rotations  of  the  screw.  A  large  divided  circle  at  one  end 
enables  fractional  parts  of  a  complete  revolution  to  be  measured. 


Fig.  5. 


In  using  the  engine,  the  object  whose  length  is  to  be 
measured  is  placed  upon  the  massive  support  underneath  the 
reading  microscope,  and  in  such  a  position  that  the  line  to  be 
measured  is  parallel  with  the  screw.  The  microscope  is  then 
moved  until  the  intersection  of  the  cross-hairs  is  directly  above 
one  end  of  this  line.  After  the  reading  of  the  divided  circle 
has  been  recorded,  the  microscope  is  again  moved  until  the 
other  end  of  the  line  to  be  measured  lies  directly  below  the 
cross-hairs.  From  the  number  of  turns  of  the  screw  necessary 
to  accomplish  this  the  length  is  computed.  The  chief  source 
of  inaccuracy  in  the  use  of  the  dividing  engine  is  the  "lost 
motion  "  between  nut  and  screw.  To  avoid  errors  arising  from 
this  source,  the  screw  should  be  turned  during  each  measure- 
ment always  in  the  same  direction.  If  the  microscope  has  by 
accident  been  carried  too  far,  do  not  attempt  to  correct  this  by 


32  JUNIOR   COURSE   IN    GENERAL   PHYSICS. 

moving  the  carriage  backwards,  but  begin  the  measurement 
again. 

No  matter  how  carefully  the  screw  of  a  dividing  engine 
has  been  cut,  it  is  impossible  to  obtain  one  that  is  perfect. 
For  the  most  accurate  determinations  the  screw  must  therefore 
be  calibrated ;  i.e.  the  pitch  must  be  determined  at  different 
points  along  the  screw  by  comparison  with  a  standard  scale. 
In  most  of  the  experiments  which  follow,  the  results  will, 
however,  be  sufficiently  accurate  if  the  errors  of  the  screw  are 
neglected.* 

The  method  of  the  experiment  is  as  follows  : 

(1)  Invert   the   thermometer,    and   allow  a  portion  of   the 
mercury  to  run  into  the  bulb  at  the  upper  end  of  the  tube. 

(2)  Separate   from   the   rest   a   thread   of    mercury   whose 
length  is  about  one-tenth  as  great  as  that  of  the  whole  tube. 
Assuming  that  a  40°  thermometer  is  used,  this  thread  will  be 
about  4°  long. 

(3)  Let  the  end  of  this  thread  be  at  the  40°  mark,  and 
measure  its  length  on  the   dividing    engine ;   then    by   gently 
jarring  the  thermometer  while  in  a  slanting  position,  move  the 
thread  until  the  end  that  was  at  40°  arrives  at  36°.     Having 
measured  its  length  in  this  position,  move  the  thread  through 
another  4°,  and  continue  in  this  way  until  it  has  reached  the 
bottom  of  the  tube. 

A  curve  is  to  be  platted  from  the  results  of  these  measure- 
ments in  which  the  positions  of  the  middle  point  of  the  thread 
are  used  as  abscissas,  and  the  reciprocals  of  its  length  as  ordi- 
nates.  The  reciprocal  of  the  length  of  the  thread  is  obviously 
proportional  to  the  average  cross-section  of  the  tube  at  the 
place  where  the  thread  is  measured.  This  curve,  therefore, 
shows  the  relative  values  of  the  cross-section  at  different  points 
along  the  tube.  Fig.  6  (I)  shows  such  a  curve. 


*  For   more    detailed    description   of    the    dividing   engine,    see   Anthony   and 
Brackett;   also  Stewart  and  Gee,  vol.  I. 


LENGTH,   TIME,   AND   MASS. 


33 


(4)  Measure  the  length  of  each  four-degree  space  on  the 
dividing  engine.  The  product  of  the  length  of  a  four-degree 
space  by  the  reciprocal  of  the  length  of  the  thread  of  mercury 
in  this  space  will  be  a  quantity  proportional  to  the  volume  of 
the  space.  If  the  thermometer  is  accurately  graduated,  this 
product  should  be  a  constant  in  all  parts  of  the  tube.  Fig.  6 
(II)  shows  the  actual  result  attained  in  the  graduation  of  a  fine 
thermometer. 

From  the  above  measurements,  the  error  in  graduation  at 
any  point  may  be  determined  in  the  following  manner :  Suppose 


.370 
.360 
.350 
.340 


~    I  Cross-section  of  bore 


.970 
.960 


-.01° 


-     II  Relative  volumes  of  4°  Spaces:     -j-5- 

ln 


12°          16°  20°  24°          28° 

CALIBRATION  CURVES  OF  A  THERMOMETER 

Fig.  6. 


32° 


36° 


that  the  range  of  the  thermometer  is  from  o°  to  40°.  Let  v 
be  the  volume  of  a  thread  of  mercury  which  is  very  nearly 
equal  in  length  to  a  four-degree  space.  Let  /x,  /2,  •••,  /10  be  the 
measured  length  of  this  thread  when  its  mid-point  is  at  the 
two-degree  mark,  six-degree  mark,  and  so  on.  Let  Llf  L2,  •••,  Z10 
be  the  measured  length  of  ist,  2d,  •••,  loth  four-degree  space. 

len  we  shall  have  for  the  mean  cross-section  of  the  «th  four- 

Jgree  space 

Sn  =  j>  (30) 

VOL.  I  —  I ) 


34  JUNIOR   COURSE   IN   GENERAL   PHYSICS, 

and  for  the  volume  of  the  nth  four-degree  space 

r.  =  v-j*  (3D 

Now  the  error  of  any  graduation  is  a  cumulative  one;  i.e.  it 
depends  on  all  the  errors  preceding  it.  The  volume  up  to  the 
end  of  the  nth  four-degree  space  is 


=  ^2^,  (32) 

^n 

and  the  total  volume  is 


The  volume  up  to  the  end  of  the  nth  four-degree  space  should 

be  —  of  the  whole  volume.     Therefore  the  volume  error  at  the 

10 
end  of  the  nth  four-degree  space  is 

E^™\L        «L  (33) 

10  1  4       i 


The  error  in  degrees  will  be 

£  L 


•*  ~r~ 
10  1  4 


Finally  a  curve  should  be  platted  (See  Fig.  6,  III),  with  values 
of  en  as  ordinates,  and  with  corresponding  values  of  n  as  abscissas. 

EXPERIMENT  A4.  Determinations  of  volumes  and  densities 
of  solids  by  measurement  of  their  dimensions. 

I. 

Determination  of  the  volume  of  a  regular  solid  by  measure- 
ment of  its  dimensions. 

If  the  solid  is  a  parallelepiped,  measure  each  of  its  twelve 
edges  on  the  dividing  engine.  If  it  is  a  cylinder,  measure  its 
altitude  in  four  places,  and  measure  the  diameter  of  each 
base  in  four  different  places.  In  each  case  great  care  should 


LENGTH,   TIME,   AND   MASS. 


35 


be  taken  that  the  microscope  moves  parallel  to  the  line  measured. 
From  the  data  obtained  compute  the  volume.  If  the  solid 
proves  to  be  pyramidal  or  conical,  treat  it  as  a  frustum.  As  a 
check  upon  the  result,  weigh  the  solid  in  air  and  in  water.  The 
difference  of  these  weights,  in  grams,  is  numerically  equal  to 
the  mass  of  the  displaced  water,  and  this  quantity  divided  by 
the  density  of  water  at  the  observed  temperature  will  give  the 
volume  of  the  solid.  In  weighing  in  water,  free  the  solid  from 
air  bubbles,  and  correct  for  the  weight  of  the  suspending  wire. 
More  accurate  results  may  be  obtained  by  correcting  for  the 
buoyancy  of  the  air.  (See  Exp.  G3.) 

II. 

Determination  of  the  volume  and  density  of  a  wire,  from 
measurements  of  length,  diameter,  etc. 

If  the  wire  is  insulated,  it  should  first  be  carefully  stripped  in 
such  a  way  as  not  to  scratch  the  surface  or  change  the  shape  of 
the  cross-section.  Then  measure  the  diameter,  at  ten  or  twelve 
different  points  throughout  the  length,  with  a  micrometer  wire 
gauge.  Before  using  the  micrometer,  its  zero  point  should  be 
tested ;  if  it  is  found  to  be  incorrect,  a  suitable  correction  must 
be  made  to  each  reading.  Measure  the  length  of  the  wire  as 
accurately  as  possible,  and  compute  its  volume,  treating  it  as  a 
cylinder  whose  diameter  is  the  mean  of  the  diameters  measured. 

(If,  however,  the  diameter  is  found  to  decrease  progressively 
from  one  end  to  the  other,  the  wire  should  be  treated  as  the 
frustum  of  a  cone.) 

Finally,  weigh  the  wire  and  compute  its  density.  Check 
the  last  result  by  determining  the  specific  gravity  by  weighing 
in  water.  (See  Exp.  Gr) 

III. 

Measurement  of  the  diameter  of  a  zvire  by  the  microscope, 
and  determination  of  density  from  diameter,  length,  and  mass. 

First  determine  the  value  in  millimeters  of  one  division  of 
the  micrometer  eye-piece.  To  do  this,  focus  the  microscope  on 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


an  accurate  scale,  and  observe  how  many  divisions  of  the  scale 
are  covered  by  any  convenient  number  of  micrometer  divisions. 

Measure  the  diameter  of  the  wire  at  ten  or  twelve  different 
points,  by  means  of  the  micrometer  eye-piece,  and  then  compute 
the  volume  and  density  of  the  wire  as  in  II,  above. 

Check  the  result  by  finding  the  specific  gravity  as  directed 
in  Exp.  Gj. 

EXPERIMENT  A5.  Determination  of  the  time  of  a  periodic 
motion  by  the  method  of  middle  elongations. 

The  method  illustrated  by  this  experiment  affords  a  means 
of  determining  the  vibration  period  of  any  vibrating  body  with 
great  accuracy.  It  is  used,  for  example,  in 
determining  the  time  of  vibration  of  the  sus- 
pended magnet  of  a  magnetometer,  in  deter- 
mining the  period  of  a  pendulum,  etc.  With 
the  object  of  affording  practice  in  the  use  of 
the  method,  the  apparatus  is  arranged  as  de- 
scribed below  : 

A  heavy  disk  (Fig.  7),  having  a  black  spot 
or  a  pencil  line  on  the  edge,  is  suspended  by  a 
long  wire,  and  is  kept  in  vibration,  when  once 
started,  by  the  torsion  of  the  wire.  Place  a 
telescope  in  some  convenient  position  near  a 
clock,  and  adjust  it  so  that  the  vertical  cross- 
hair is  in  the  prolongation  of  the  wire.  The 
black  spot  will  then  move  back  and  forth  in 
the  field,  passing  the  cross-hair  twice  in  each 
vibration.  Note  the  time  of  day  (hour,  minute, 
second,  and  tenth  of  a  second)  of  each  passage 
of  the  spot  across  the  hair,  for  ten  successive 
transits.  To  obtain  the  time  accurately,  ol> 
Fig.  7.  -Disk  for  Tor-  serve  the  second  hand  of  the  clock  and  count 

sional  Vibrations. 

seconds  as  indicated  by  it.     Continue  the  count 
while  observing  the  transit,  looking  occasionally  at  the  clock  to 


LENGTH,   TIME,   AND   MASS.  37 

see  that  no  mistake  is  made.  In  most  cases  the  time  of  transit  will 
not  correspond  exactly  to  the  beginning  of  a  second.  Observe 
the  position  of  the  spot  at  the  second  just  before,  and  again  at 
the  second  just  after  the  transit :  from  the  relative  distances  of 
these  two  positions  from  the  cross-hair  the  tenths  of  a  second 
can  be  estimated.  This  will  doubtless  at  first  be  somewhat  dif- 
ficult, but  after  a  little  practice  the  estimation  can  be  made  with 
considerable  accuracy.  An  experienced  observer  should  be  able 
to  estimate  twentieths  of  a  second  with  certainty.  Repeat  the 
ten  readings  mentioned  above  at  intervals  of  about  fifteen  min- 
utes until  three  sets  have  been  taken  of  ten  observations  each. 

To  utilize  these  data  in  computing  the  period  in  question, 
add  together  the  fifth  and  sixth  time  of  transit  in  each  set  and 
divide  by  two.  The  result  will  be  the  time  of  the  "  Middle 
Elongation,"  or  the  time  at  which  the  spot  was  at  its  greatest 
distance  from  the  cross-hair  between  the  fifth  and  sixth  transits. 
If  all  the  observations  were  correct,  the  same  time  of  middle 
elongation  would  be  found  by  adding  together  the  fourth  and 
seventh,  the  third  and  eighth,  etc.,  and  in  each  case  dividing  by 
two.  In  general,  however,  the  five  values  obtained  for  the  time 
of  middle  elongation  will  differ  slightly  on  account  of  errors  in 
the  observations,  and  their  average  should  be  used.  Subtract- 
ing the  time  of  one  middle  elongation  from  that  of  the  next,  and 
dividing  by  the  number  of  vibrations  in  the  interval,  gives  the 
time  of  vibration  with  great  accuracy.  It  is  not  necessary  to 
count  the  vibrations ;  the  number  may  be  deduced  from  the 
observations  themselves.  Between  the  first  and  ninth,  or  second 
and  tenth  observations  of  each  set,  there  were  four  vibrations. 
Dividing  the  interval  between  the  first  and  ninth  observations 
by  four,  gives  an  approximation  to  the  periodic  time.  If  the 
interval  between  two  middle  elongations  is  divided  by  this  quan- 
tity, the  quotient  would,  if  the  observations  were  all  exact,  be  a 
whole  number ;  *  i.e.  the  number  of  vibrations  in  the  interval. 

*  It  is  to  be  observed  that  this  quotient  might  also  be  a  whole  number  plus  a 
half.     This  will  be  the  case  if  the  disk  moved  in  opposite  directions  at  the  beginning 
the  two  sets  of  observations. 


38  JUNIOR   COURSE   IN    GENERAL   PHYSICS. 

It  should,  with  reasonably  accurate  observations,  be  near  enough 
to  a  whole  number  to  leave  no  doubt  as  to  the  true  number  of 
vibrations.  Dividing  the  interval  by  this  number  gives  the 
periodic  time  desired.  As  a  check,  the  time  of  vibration  should 
also  be  computed  from  the  interval  between  the  second  and 
third  middle  elongations. 

It  is  to  be  observed  that  the  interval  which  it  is  safe  to  allow 
between  two  sets  of  observations  depends  upon  the  accuracy  of 
the  observations,  and  upon  the  length  of  the  period  to  be  deter- 
mined. If  the  period  is  short,  the  interval  between  two  sets  of 
observations  must  also  be  short.  Determine,  from  a  comparison 
of  your  observations,  how  long  an  interval  would  have  been  safe. 


.   / 

/^                       <f 

M 

N 

\ 
\ 

\6' 

A.  ----- 

\                            / 

"/ 

v      r 
\J 

\  7 

\    / 

Fig.  8. 

Repeat  the  experiment  with  the  cross-hair  to  one  side  of  the 
center,  and  show  that  the  method  pursued  eliminates  any  such 
want  of  symmetry. 

The  principle  of  this  method  may  perhaps  be  more  clearly 
understood  if  the  motion  of  the  disk  is  represented  graphically, 
as  in  Fig.  8.  Horizontal  distances  here  represent  times,  while 
vertical  distances  correspond  to  the  displacement  of  the  disk 
from  its  middle  position. 

The  sinuous  line  in  Fig.  8  thus  represents  graphically  the 
displacement  of  the  disk  as  a  function  of  the  time.  The  pas- 
sage of  the  spot  across  the  cross-hairs  of  the  telescope  corre- 
sponds in  the  figure  with  the  intersection  of  the  curve  with  the 
line  A'B' .  When  the  cross-hairs  are  placed  in  the  prolongation 
of  the  suspending  wire,  this  line  coincides  with  the  middle  line 
AB.  In  general  it  is  displaced  as  shown.  The  time  of  middle 


LENGTH,    TIME,   AND   MASS. 


39 


elongation  corresponds  to  the  point  M  on  the  curve,  and  lies 
midway  between  the  times  5  and  6,  4  and  7,  etc.  It  is  evident 
also  that  M  lies  midway  between  5'  and  6',  4'  and  f,  etc.  In 
other  words,  the  method  is  independent  of  the  position  of  the 
cross-hairs.  Since  M  corresponds  to  the  time  at  which  the 
vibrating  body  was  at  rest,  it  is  clear  that  the  time  of  middle 
elongation  is  independent  of  the  position  of  the  telescope.  A 
movement  of  the  latter  between  two  sets  of  observations  is 
therefore  without  effect  on  the  result. 

When  the  time  of  vibration  is  less  than  four  or  five  seconds, 
the  observations  become  difficult,  and  in  such  cases  an  electrical 
contact  is  provided  by  means  of  which  the  successive  transits 
are  automatically  recorded  upon  a  chronograph.  The  principle 
of  the  method  remains,  however,  unaltered. 

As  an  example  of  the  employment  of  the  method,  the  fol- 
lowing set  of  observations  is  appended  : 

DATE:  JAN.  4,  1894. 


First  Set. 

Second  Set. 

No. 

h. 

m. 

sec. 

No. 

h. 

m.     sec. 

I  ... 

•3: 

14: 

IO.2 

I.  .  . 

-3: 

35:   9-3 

2.  .  . 

3  = 

14: 

23-7 

Middle 

Elongations. 

2..  . 

•3  = 

35  :  22.8 

Middle  Elongations. 

3  .- 

.3: 

14: 

35-9 

5-6... 

3^5 

:8.25 

3... 

•3: 

35  :  35.° 

5-6.... 

3:36:745 

4... 

3  = 

14: 

49-3 

4-7... 

.3:  15 

:8.25 

4... 

•3: 

35  :  48.4 

4-7.  ... 

3:36:74 

$••• 

.3: 

15: 

1.5 

3-8... 

.3:15 

:8.3o 

5". 

•3: 

36:    0.7 

3-8..-. 

3:36:74 

6... 

3: 

15: 

15.0 

2-9... 

.3:  15 

=  8.35 

6... 

•3: 

36:14.2 

2-9.... 

3:36:745 

7  ... 

3  = 

15: 

27.2 

I-IO.  .. 

.3:  15 

=  8.30 

7... 

•3: 

36:26.4 

I-IO.  .  .  . 

3:36:745 

8... 

3: 

15: 

40.7 

Average, 

3:15 

:8.29 

.  8... 

•3: 

36:39.8 

Average, 

3:36:743 

9... 

3: 

15: 

53-0 

9... 

-3: 

36:52.1 

10..  . 

3: 

16: 

6-4 

10.  .  . 

•3: 

37:   5-6 

Interval  between  first  and  second  middle  elongations  =  20  m. 
59.14  second  =  1259.14  second. 

Approximate  time  of  one  vibration  computed  from  interval 
between  ist  and  9th  observation  of  first  set  =  25.7  second. 

— •*"'  ^  =  48.9+  ;  nearest  whole  number  =  49. 
25.7 

...  Period  =I259-I4=  25.697. 
49 

It  is  propable  that  the  result  is  correct  to  within  a  unit  in 
the  third  place  of  decimals. 


40 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


GROUP  B:    STATICS. 

(Bx)  Parallelogram  of  forces  ;    (B2)  Parallel  forces  ;    (B3)  Prin- 
ciple of  moments. 

EXPERIMENT  Br     The  parallelogram  of  forces. 

To  illustrate  the  principle  of  the  "parallelogram  of  forces," 
the  simple  apparatus  shown  in  Fig.  9  may  be  employed.  It 
consists  of  a  circular  board  about  a  meter  in  diameter,  which  is 
held  in  a  vertical  plane  by  any  suitable  support.  Cords,  joined 


Fig.  9.  —  Circular  Blackboard  for  the  Study  of  the  Parallelogram  of  Forces. 

together  at  the  center  of  the  board,  may  be  made  to  take  any 
desired  directions  by  passing  over  adjustable  pulleys  Plf  P2,  etc. 
By  properly  adjusting  the  position  of  these  pulleys,  and  by 
hanging  suitable  weights  on  the  cords,  it  is  possible  to  obtain 
any  desired  system  of  forces  acting  at  the  point  C.  If  the 
system  of  forces  is  in  equilibrium,  the  point  of  intersection 
of  the  cords  will  not  move,  even  when  free  to  do  so.  The  ap- 


STATICS.  41 

paratus  thus  affords  a  means  of  testing  roughly  the  solutions  to 
problems  which  involve  the  equilibrium  of  forces  acting  in  one 
plane  and  applied  at  a  single  point. 

The  student  should  test  in  this  way  the  solutions  to  three 
problems  such  as  those  given  below  :  * 

1.  Two  forces  being  given,  together  with  the  angle  between 
them,  their  resultant  is  to  be  computed  both  in  magnitude  and 
direction. 

2.  Three  forces  being  given,  compute  the  angles  that  they 
must  make  with  each  other  in  order  to  be  in  equilibrium. 

3.  Four  forces  are  given  in  magnitude  and  direction.     Com- 
pute their  resultant. 

In  the  last  problem,  the  simplest  method  of  solution  is  to 
resolve  each  force  into  two  components  along  arbitrary  rec- 
tangular axes,  f 

In  testing  the  results,  it  will  be  found  most  convenient  to 
fasten  the  point  of  intersection  at  the  center  of  the  board  until 
the  weights  have  all  been  applied.  If,  when  the  cords  are  now 
left  free  to  move,  their  point  of  intersection  still  remains  at  the 
center,  it  is  clear  that  the  various  forces  are  in  equilibrium,  and 
that  the  solution  of  the  problem  is  reached.  In  the  first  prob- 
lem, for  example,  the  two  given  forces  should  be  in  equilibrium 
with  the  equal  and  opposite  of  their  resultant.  To  show  that 
the  equilibrium  is  not  due  to  friction,  the  junction  of  the  cords 
may  be  displaced  from  the  center  ;  it  should  then  vibrate  back 
and  forth  about  its  position  of  equilibrium,  and  finally  come  to 
rest  not  far  from  the  center. 

In  the  report  on  this  experiment  the  dependence  of  the 
computations  upon  the  principle  of  the  parallelogram  of  forces 
should  be  clearly  explained  in  each  of  the  three  cases.  Dia- 
grams should  also  be  drawn  showing  the  exact  position  in  which 
the  weights  were  placed  when  testing  the  results. 

*  Numerical  data  will  be  furnished  by  the  instructor, 
f  See  Church's  Mechanics. 


42  JUNIOR   COURSE    IN    GENERAL   PHYSICS. 

EXPERIMENT  B2.     Parallel  forces. 

The  apparatus  used  in  this  experiment  consists  of  a  hori- 
zontal graduated  bar,  whose  weight  may  be  counterbalanced. 
At  any  point  along  the  bar  weights  may  be  suspended  by  means 
of  stirrups ;  while  forces  may  be  made  to  act  upwards  on  the 
bar  by  means  of  short  levers,  which  can  be  attached  at  any  point 
desired.  Three  or  more  problems  involving  the  equilibrium  of 
parallel  forces  are  to  be  given  to  the  student,  and  the  experiment 
consists  in  verifying  the  results  that  are  obtained  by  computa- 
tion. In  the  first  problem  two  forces  and  their  points  of  appli- 
cation are  given,  to  determine  their  resultant  in  position  and 
magnitude.  In  the  second  problem  three  or  more  forces  acting 
in  the  same  direction  are  given,  together  with  their  points  of 
application,  to  determine  the  resultant.  In  the  third  problem 
three  or  more  forces  acting  in  different  directions  are  given, 
and  the  resultant  is  required  as  before. 

In  the  report  on  this  experiment  the  method  of  working  the 
problems  should  be  fully  explained.  Diagrams  should  also  be 
given  showing  the  position  and  magnitude  of  the  weights  actually 
used  in  testing  the  results. 

EXPERIMENT  B3.     Principle  of  moments. 

The  apparatus  used  in  this  experiment  consists  of  a  light 
frame  which  is  free  to  revolve  about  an  axis  placed  at  the  center 


d 

i 

b 

L 

Jl 

d 

u 

Fig.  10.  — Apparatus  for  the  Study  of  Moments. 

of  a  rectangular  table  (Fig.  10).     Cords  passing  over  pulleys  on 
the  edge  of  the  table  may  be  attached  to  the  frame  at  various 


STATICS. 


43 


points  by  means  of  hooks  or  pins  (see  Fig.  n).  By  hanging 
suitable  weights  at  the  ends  of  these  cords  and  properly  adjust- 
ing the  position  of  the  pulleys,  it  is  therefore  possible  to  obtain 
forces  acting  on  the  frame,  whose  magnitude  and  direction  are 
under  control. 


/    ^ 

V 

r 

\ 

X" 

> 

A 

"\ 

\ 

A 

&* 

TAN 

\ 

^ 

^ 

b 

\tf 

a 

\ 

\ 

Fig.   11. 

The  object  of  the  experiment  is  to  verify  by  means  of  this 
apparatus  the  solutions  of  problems  involving  the  principle  of 
moments. 

1.  Four  forces  are  given  in  direction,  magnitude,  and  point 
of  application.     It  is  required  to  find  the  magnitude  of  a  fifth 
force  which  will  produce  equilibrium  when  applied  at  a  given 
point  and  acting  in  a  given  direction. 

2.  Four  forces  are  given  as  above,  together  with  the  magni- 
tude and  point  of  application  of  a  fifth  force.     It  is  required  to 
find  the  direction  in  which  this  last  force  must  act  in  order  to 
produce  equilibrium. 

To  verify  the  results  obtained  by  calculation,  fasten  the 
frame  by  means  of  a  pin  in  one  other  point  besides  the  axis,  so 
that  it  is  no  longer  free  to  revolve.  Then  attach  the  cords  to 


44 


JUNIOR   COURSE   IN    GENERAL   PHYSICS. 


the  frame,  and  adjust  the  position  of  the  pulleys,  so  that  the 
forces  to  which  the  frame  is  subjected  are  as  given  in  the 
problem.  Adjust  in  like  manner  the  direction  and  intensity  of 
the  force  which  is  to  produce  equilibrium.  Now  remove  the 
pin,  so  that  the  frame  is  again  free  to  rotate.  If  the  computa- 
tions are  correct,  the  position  of  the  frame  should  remain  unal- 
tered. To  prove  that  the  frame  is  not  held  in  position  by 
friction,  it  may  be  displaced  through  a  small  angle  by  the  hand. 
The  action  of  the  various  forces  will  then  cause  it  to  return, 
after  a  few  vibrations,  to  its  original  position. 

In  the  report  on  this  experiment  the  method  of  working  the 
problems  should  be  clearly  explained.  Diagrams  of  the  appa- 
ratus should  also  be  given,  in  which  are  shown  the  directions  and 
magnitudes  of  the  forces  actually  used  in  each  test. 


GROUP  C:   FRICTION  AND  SIMPLE   MACHINES. 

Coefficient   of  friction;    (C2)    Law    of   wheel   and    axle ; 
(C3)  Law  of  systems  of  pulleys. 

EXPERIMENT  Cr     To  determine  the  coefficient  of   friction 
between  two  surfaces. 


w 


J 


1                                                \  \\J. 

\ 

Fig.  12.  — Coefficient  of  Friction. 

The  apparatus,  which  is  shown  in  Fig.  12,  consists  (i)  of  a 
smooth  plate  made  of  one  of  the  materials  to  be  tested  and 
capable  of  being  adjusted  so  that  its  upper  surface  is  accu- 


FRICTION   AND    SIMPLE   MACHINES. 


45 


rately  horizontal  ;  (2)  a  small  block  of  the  second  material  in 
question  which  can  be  made  to  slide  across  the  plate  by  means 
of  a  cord  passing  over  a  pulley  and  loaded  with  suitable  weights. 

Observations  should  be  taken  as  follows  : 

First  adjust  the  plate  so  that  its  surface  is  horizontal.  Place 
the  block  upon  it,  and  add  enough  weights  to  make  the  total 
pressure  five  kilograms.  Then  hang  weights  on  the  cord  until 
the  force  is  just  sufficient  to  keerj  the  block  moving  uniformly 
when  once  started.  Repeat  the  observations  with  pressures  of 
10,  15,  20,  etc.,  kilos  on  the  block  until  a  pressure  of  50  kilos  is 
reached. 

It  is  to  be  observed  that  the  weights  upon  the  cord  do 
not  represent  exactly  the  force  required  to  overcome  the 
friction  between  plate  and  block.  A  correction  must  be  applied 
in  each  case  on  account  of  the  friction  of  the  pulley  itself.  To 
determine  this  correction,  a  cord  may  be  passed  over  the  pul- 
ley, carrying  equal  weights  at  its  two  ends.  A  definite  press- 
ure is  thus  exerted  on  the  bearings  of  the  pulley,  and  to 
overcome  the  resulting  friction,  a  slight  additional  weight, 
whose  amount  is  determined  by  experiment,  must  be  placed 
on  one  side.  In  this  way  the  relation  between  the  friction  of 
the  pulley  and  the  pressure  on  its  bearings  can  be  determined, 
after  which  the  corrections  to  be  applied  to  the  former  observa- 
tions can  be  readily  computed.* 

The  results  may  now  be  best  shown  by  platting  a  curve  on 
cross-section  paper,  using  pressures,  W  (Fig.  12)  as  abscissas 
and  forces  necessary  to  overcome  friction,  w  (Fig.  12)  as  ordi- 
nates.  If  friction  is  proportional  to  pressure,  this  curve  should 
be  a  straight  line  passing  through  the  origin.  Find  its  equation 
by  the  method  of  least  squares,  and  so  deduce  the  coefficient. 
A  typical  curve  of  the  kind  described  is  shown  in  Fig.  13. 

Each  of  the  observations  at  different  pressures  should  be 

*  It  is  to  be  observed  that  the  friction  of  the  pulley  is  determined  by  the  pressiire 
on  its  bearings,  and  is  independent  of  the  direction  of  this  pressure.  The  weight  of 
the  pulley  itself  is  usually  so  small  that  it  can  be  neglected. 


46 


JUNIOR   COURSE   IN    GENERAL   PHYSICS. 


independent,  and  uninfluenced  by  any  assumption  as  to  the 
probable  result.  Friction,  under  the  best  of  conditions,  is 
irregular,  so  that  it  need  not  be  at  all  surprising  if  the  observa- 
tions are  somewhat  discordant.  The  best  final  results  will  be 
obtained  by  making  a  number  of  entirely  independent  observa- 


8.00 


i4.00 


2.00 


FRICTION 


10 


PRESSURE  (W) 
Fig.    13. 


tions,  each  one  being  as  carefully  made  as  though  it  alone  were 
to  determine  the  coefficient. 

The  same  apparatus  may  be  employed  to  determine  the 
influence  of  the  area  of  contact  upon  the  coefficient  of  friction, 
and  also  to  study  the  "friction  of  rest,"  or  "starting  friction." 

EXPERIMENT  C2.  Law  of  the  wheel  and  axle  and  deter- 
mination of  efficiency. 

In  this  experiment  a  small  weight  suspended  by  a  cord  from 
a  large  wheel  is  made  to  lift  a  larger  weight  which  hangs  from 
the  axle  of  the  wheel.*  The  object  of  the  observations  is  to 

*  The  experiment  will  perhaps  be  more  instructive  if  a  compound  wheel  and  axle 
is  used,  or  a  compound  system  consisting  of  an  endless  screw  and  gear  wheel.  In 
these  cases  the  influence  of  friction  on  the  results  will  be  much  more  marked. 


FRICTION    AND    SIMPLE   MACHINES. 


47 


determine  experimentally  the  relation  between  the  two  weights 
when  the  smaller  is  just  sufficient  to  keep  the  system  moving. 
It  is  to  be  observed  that  the  conditions  differ  from  those  con- 
sidered in  the  simple  theory  of  the  wheel  and  axle,  in  the  fact 
that  the  friction  of  the  various  parts  is  not  negligible.  The 
system  forms,  in  fact,  a  simple  type  of  machine,  whose  object 
we  may  consider  to  be  the  raising  of  weights.  The  effect  of 
friction  in  reducing  the  efficiency  of  this  simple  machine  is 
exactly  the  same  in  kind  as  it  is  in  larger  and  more  complicated 
machines,  and  the  experiment  thus  affords  an  opportunity  of 
studying  the  influence  of  friction  in  a  simple  case  where  the 
various  disturbing  factors  may  be  readily  isolated. 

Observations  are  to  be  taken  as  follows  : 

Find  by  experiment  the  weights  necessary  to  raise  loads  of 
5,  10,  15,  up  to  50  kilos,  the  small  weight  being  adjusted  in 


20  30 

LOAD 

Fig.  14. 

each  case  until  it  is  just  sufficient  to  keep  the  system  moving 
with  a  slow,  uniform  motion,  when  started  by  the  hand.     Make 
several  trials  with  each  load  and  use  the  mean  of  the  results. 
It  is  essential  that  each  observation  should  be  entirely  inde- 
mdent  of  all  the  rest,  and  uninfluenced  by  any  assumption  as 
what  the  relation  should  be  between  "power"  and  "load." 
From  the  data  thus  obtained,  plat  curves  showing  the  rela- 
ion  between  the  power  and  the  load  in  each  case.     Fig.  14 
lows  such  a  curve.     To  locate  points  on  these  curves  (which 


48 


JUNIOR   COURSE  .IN   GENERAL   PHYSICS. 


should  be  accurately  drawn  on  cross-section  paper),  the  loads 
are  to  be  used  as  abscissas  and  the  corresponding  powers  as 
ordinates.*  From  the  appearance  of  the  curves  decide  upon 
the  form  of  their  equations,  and  find  the  constants  by  the 
method  of  least  squares.  The  lines  represented  by  the  equa- 
tions that  are  obtained  by  least  squares  should  be  drawn  on  the 


.90 


.80 


.70 


WHEEL 


AND  AXLE 


CURVE 


10 


20 


30 
LOAD 

Fig.  15. 


40 


50 


same  sheet  as  the  original  ones,  in  order  to  see  how  closely  they 
represent  the  observations. 

Having  determined  the  velocity  ratio  in  each  case,  show 
what  the  behavior  of  the  apparatus  would  be  if  there  were  no 
friction,  and  compute  the  efficiency  of  the  apparatus,  considered 
as  a  machine  for  lifting  weights,  for  loads  of  5,  10,  25,  and  50 
kilos.  A  curve  showing  the  relation  between  efficiency  and 
load  may  then  be  drawn  (see  Fig.  15). 

The  velocity  ratio  may  be  roughly  computed  from  the 
diameters  of  the  wheel  and  axle ;  but  on  account  of  the  appre- 
ciable thickness  of  the  rope  used,  it  is  better  to  obtain  the 
velocity  ratio  by  actually  measuring  the  distance  passed  over 
by  the  load  when  the  wheel  is  turned  a  known  number  of  times. 


*  Note  that  the  horizontal   and   vertical   scales  need   not   be   the   same.     See 
Introduction. 


FRICTION   AND   SIMPLE   MACHINES.  49 

Addenda  to  the  report : 

(1)  Interpret  the  curves  obtained  in  detail.     For  example, 
the  friction  of  the  machine  consists  of  two  parts  :  (i)  a  constant 
frictional  resistance,  which  is   independent  of   the  load;    (2)  a 
variable    resistance   becoming    greater    as    the   load   increases. 
Each  of  these  is  readily  determined  from  the  curve. 

(2)  Indicate  the  greatest    possible    efficiency  that   can    be 
attained  by  the  machine,  and  the  load  to  which  this  corresponds. 

EXPERIMENT  C3.  To  determine  the  efficiency  of  a  system  of 
pulleys. 

In  this  experiment  a  system  of  pulleys  is  used  by  which  a 
small  weight  moving  through  a  considerable  distance  is  enabled 
to  lift  a  much  larger  weight  through  a  comparatively  small 
distance.  The  objects  of  the  experiment  are  :  (i)  To  determine 
experimentally  the  relation  between  "power"  and  "load "for 
uniform  motion ;  (2)  to  determine  the  efficiency  of  the  system 
considered  as  a  machine  for  raising  weights.  The  procedure  is 
as  follows  : 

(1)  Find  by  experiment  the  weights  necessary  to  raise  loads 
of  0.5,  i.o,  1.5  ,  up  to  6  kilos,  the  small  weight  being  adjusted  in 
each  case  until  it  is  just  sufficient  to  maintain  uniform  motion 
when  the  system  is  started  by  the  hand.     Make  several  trials 
with  each  load,  and  use  the  mean  of  the  results. 

(2)  With  the  data  obtained,  plat  curves  showing  the  relation 
between  power  and  load  for  both  cases,  and  from  the  appear- 
ance of  the  curves  decide  upon  the  form  of  their  equations. 
The  constants  are  to  be  determined  by  the  method  of  least 
squares.     The  lines  represented  by  the  equations  obtained  by 
least  squares  should  be  drawn  on  the  same  sheet  as  the  original 
curves. 

(3)  Having  determined  the  ratio  of  the  distances  passed  over 
by  the  two  weights,  show  what  powers  would  be  necessary  to 
raise  the  same  loads  if  there  were  no  friction,  and  compute  the 

VOL.  I  —  E 


50  JUNIOR   COURSE    IN    GENERAL   PHYSICS. 

efficiencies  of   the   two  systems  for   loads  of   0.5,    i,   3,  and  6 
kilos. 

The  results  are  to  be  discussed  in  the  manner  explained  in 
the  previous  experiment. 


GROUP  D:    UNIFORMLY  ACCELERATED  MOTION. 

(Dj)    Atwood's  machine.     (D2)   Determination  of  gravity  from 
the  motion  of  a  freely  falling  body. 

EXPERIMENT  Dr     Atwood's  machine. 

In  Atwood's  machine  a  vertical  standard,  from  two  to  three 
meters  high,  carries  at  the  top  a  light  pulley,  P  (Fig.  16),  which 
is  mounted  in  such  a  way  as  to  make  the 
friction  of  its  bearings  as  small  as  possible. 
To  the  standard  is  attached  a  scale  grad- 
uated in  centimeters  or  inches  for  conven- 
ience in  measurement.  Over  the  pulley 
hangs  a  light  silken  cord,  to  which  weights, 
wlt  wv  may  be  hung.  If  equal  weights 
are  hung  on  the  two  sides  of  the  pulley,  it 
is  evident  that  the  system  will  remain  at 
rest.  But  if  a  small  additional  weight  be 
placed  on  one  side,  the  condition  of  equi- 
librium will  be  destroyed,  and  the  heavier 
side  will  begin  to  fall  with  a  uniformly 
accelerated  motion.  The  force  of  gravity 
acting  on  the  small  added  mass,  or  "rider," 
*- (Fig.  17),  is  thus  utilized  to  set  in  motion 
a  much  larger  mass,  and  the  acceleration 
is,  in  consequence,  smaller  than  if  the  rider 
alone  were  moved.  By  suitably  choosing 
16  the  various  weights,  the  motion  may  be 

made  so  slow  that  the  velocity  can  be  readily  measured.     The 
apparatus  thus  affords  a  means  of  illustrating  the  laws  of  uni- 


UNIFORMLY   ACCELERATED   MOTION. 


formly  accelerated  motion,  and  can  also  be  used,  as  explained 
below,  to  determine  the  acceleration  of  gravity,  g. 

For  convenience  in  measuring  time,  most  forms  of  Atwood's 
machine  are  provided  with  an  electric  bell  or  sounder,  which  can 
be  connected  with  a  seconds  pendulum.  By  ; 
means  of  an  electromagnet,  m  (Fig.  16),  placed 
at  the  top  of  the  vertical  standard,  and  con- 
nected with  the  same  circuit  as  the  sounder, 
the  weights  may  be  released  exactly  at  the 
beginning  of  a  second,  so  that  the  necessity 
of  estimating  fractions  of  a  second  is  avoided. 
A  bracket,  ss  (Fig.  16),  movable  along  the 
upright  standard,  may  be  adjusted  so  as  to 
stop  the  fall  at  any  point  desired,  while  a 
ring,  s2,  also  adjustable  in  position,  serves  to 
remove  the  rider  at  any  desired  time  without 
disturbing  the  motion  of  the  weights  them- 
selves. 


Fig.  17. 


I. 


To  test  the  laws  of  uniformly  accelerated  motion. 

Hang  equal  weights  on  the  two  sides  of  the  pulley,  and  then 
put  enough  additional  weight  on  the  side  which  is  to  fall  during 
the  experiment  to  overcome  the  friction  of  the  apparatus.  This 
can  be  done  by  adding  small  pieces  of  paper  or  tin-foil  until  the 
weight  will  continue  to  move  uniformly  downward  when  once 
started.  When  this  adjustment  is  completed,  place  the  rider  in 
position,  and  adjust  the  ring  by  trial  to  such  a  position  on  the 
vertical  bar  that  it  will  remove  the  added  weight  after  a  fall  of 
exactly  two  seconds.  Measure  the  distance  traversed  by  the 
rider  and  record  it,  together  with  the  time  of  fall.  To  deter- 
mine the  velocity  acquired,  adjust  the  bracket  to  such  a  position 
that  the  space  between  it  and  the  ring  shall  be  traversed  in 
some  exact  number  of  seconds.  This  distance  between  ring  and 
table  being  measured,  the  velocity  can  be  computed.  To  insure 


JUNIOR   COURSE   IN   GENERAL  PHYSICS. 


accuracy,  each  of  these  observations  should  be  repeated  several 
times  and  the  average  of  the  results  used.  Now  shift  the  posi- 
tion of  the  ring  until  the  time  of  fall  is  three  seconds  ;  then  four 


L. 


ATWOOD'S  MACHINE 


20cm.       40cm.       60cm.       80cm.      100cm. 

SPACE  TRAVERSED 

Fig.   18. 


seconds ;  and  so  on,  repeating  the  observations  described  above 
in  each  case. 

The  results  can  be  best  shown  by  platting  two  curves,  one 


/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

z 

10cm.20       30       40       50       60       70       80      90      10 
VELOCITY  ACQUIRED 

Fig.   19. 


showing  the  relation  between  time  of  fall  and  the  space  trav- 
ersed (Fig.  1 8),  the  other  showing  the  relation  between  time  of 


UNIFORMLY   ACCELERATED   MOTION.  53 

fall  and  the  velocity  acquired  (Fig.  19).  Discuss  the  results 
and  show  whether  or  not  they  are  in  agreement  with  the  laws  of 
uniformly  accelerated  motion. 

II. 

To  use  Atwood's  machine  for  the  determination  of  g. 

If  the  mass  of  the  rider  is  m,  the  resultant  force  acting  on 
the  system  is  mg.  This  force  is  equal  to  the  product  of  the 
total  mass  moved  into  the  acceleration  imparted.  If,  therefore, 
the  total  mass  except  the  rider  be  denoted  by  M,  and  the 
measured  acceleration  by  a,  we  have 

mg=(m+M)a;  (34) 

g  can  therefore  be  computed  as  soon  as  m,  M,  and  a  are  known. 
The  mass  m  can  be  at  once  determined  by  weighing,  while  a 
can  be  computed  from  the  observations.  But  the  value  of  M 
cannot  be  so  simply  obtained,  since  the  pulley  itself  forms  a 
part  of  the  mass  set  in  motion.  The  " equivalent  mass"  of  the 
pulley  must  therefore  be  first  determined.  To  accomplish  this, 
proceed  as  follows  : 

(1)  Remove  one  of  the  small  weights  from  each  side  of  the 
cord ;  adjust  again  with  tin-foil  to  overcome  friction,  and  deter- 
mine by  experiment  the  spaces  corresponding  to  falls  of  two, 
three,  four,  etc.,  seconds,  respectively. 

(2)  Determine  also  the  mass  upon  the  cord. 

(3)  Repeat  the  observations  after  removing  another  weight 
from  each  side,  and  continue  until  only  one  weight  remains. 

From  these  observations,  compute  the  acceleration  imparted 
by  the  rider  in  each  case.  Since  the  equivalent  mass  of  the 
pulley  is  known  to  be  a  constant,  it  may  now  be  readily  com- 
puted, either  algebraically  or  graphically.  The  graphical 
•  method  which  follows  is,  however,  recommended. 

Plat  a  curve  (see  Fig.  20)  upon  cross-section  paper  in  which 
the  masses  hung  upon  the  pulley  are  used  as  abscissas,  and  the 


54 


JUNIOR   COURSE    IN   GENERAL   PHYSICS. 


reciprocals  of  the  corresponding  accelerations  as  ordinates.  This 
curve  should,  if  the  observations  are  good,  be  nearly  a  straight 
line.  The  equation  of  the  line  is,  in  fact, 


(35) 


where  M0  denotes  the  constant  equivalent  mass  of  the  pulley, 
and  m + M  the  sum  of  the  masses  hung  from  the  cord.     The 

co-ordinates  of  the  curve  are  therefore  x=  m  + M  and  jj>  =  - ;  i.e. 

a 

(36) 


RECIPROCAL  OF  ACCELERATION 

8  S  8  fe  8  8  3  S  8  ^ 

ATWi 

)OD'J 

MAC 

MINE 

/ 

/ 

/ 

/ 

X 

/ 

/ 

/ 

s 

/ 

/ 

/ 

/ 

/ 

/ 

x^ 

aoo 


300 
MASSES 

Fig.  20. 


300 


400 


500 


This  is  an  equation  of  the  first  degree,  and  therefore  repre- 
sents a  straight  line. 

Owing  to  errors  of  observation,  the  curve  obtained  will 
not  be  exactly  straight.  A  straight  line  should,  however,  be 
drawn  which  passes  as  nearly  as  possible  through  all  the  points 
platted.  A  little  consideration  will  show  that  the  intercept  of 
this  line  on  the  axis  of  abscissas  is  equal  to  the  equivalent  mass 
of  the  pulley. 

The  computation  of  g  can  now  be  easily  performed. 


UNIFORMLY   ACCELERATED   MOTION. 


55 


It  may  be  readily  proved  that  what  has  been  called  the  equiva- 
lent mass  of  the  pulley  is  really  its  moment  of  inertia  divided 
by  the  square  of  the  distance  from  its  center  to  the  cord.  The 
work  done  by  gravity  when  the  rider  has  moved  a  distance,  /, 
is  mgl,  but  this  work  must  be  equal  to  the  kinetic  energy 

gained. 

.'.  mlg=±(m+M)v*  +  ±K<J,  (37) 

• 
in  which  v  is  the  final  velocity  of  the  suspended  masses,  CD  the 

final  angular  velocity  of  the  pulley,  and  K  its  moment  of  inertia. 
Remembering  that  z/2=2#/and  v=ru>,  this  equation  reduces  to 


(38) 


in  which  r  is  the  radius  of  the  pulley. 


EXPERIMENT  D2.     Determination  of  g  from  the  motion  of  a 
freely  falling  body. 

The  apparatus  for  this  experiment,  Fig.  21,  is  so  arranged 
that  a  piece  of  smoked  glass  may  be  allowed  to  fall  freely  in 


Fig.  21. 

front  of  a  vibrating  tuning-fork  of  known  pitch.  A  stylus 
attached  to  one  prong  of  the  fork  is  adjusted  to  trace  a  sinuous 
line  on  the  glass  as  it  falls.  By  measuring  the  length  of  the 
successive  waves  of  this  curve,  it  is  possible  to  compute  the 
acceleration  of  gravity.  As  a  means  of  measuring  gt  the  method 
is  not  at  all  accurate,  since  any  friction  in  the  apparatus  will 
introduce  a  considerable  error.  The  experiment  is  valuable, 
however,  in  illustrating  the  laws  of  falling  bodies,  and  in  familiar- 


56  JUNIOR   COURSE   IN    GENERAL   PHYSICS. 

izing  the  student  with  the  use  of  the  dividing  engine  as  an  in- 
strument for  measuring  length. 

Having  covered  the  glass  with  a  thin  layer  of  smoke  (pref- 
erably from  burning  camphor),  adjust  the  stylus  until  it  traces 
a  smooth  and  distinct  curve  when  the  glass  is  allowed  to 
fall.     Several  trials  may  be  necessary  before  this  adjust- 
ment is  satisfactory.  When  a  good  curve  has  been  obtained, 
stop  the  vibration  of  the  fork,  and  allow  the  glass  to  fall  a* 
)     second  time  without  changing  the  position  of  the  glass. 

The  stylus  will  then  be  made  to  trace  a  straight  line 
)     nearly  through  the  center  of  the   sinuous  curve.     (See 
Fig.  22.) 

Now  adjust  the  glass  under  the   microscope  of   the 
)     dividing  engine,  assume  some  sharply  defined  intersection 
of  the  straight  line  and  curve  as  a  starting-point,  and 
measure  the  distance  from  this  to  the  third,  fifth,  seventh, 
)     etc.,  intersection.     These  distances  evidently  represent 
the  spaces  passed  over  during  one,  two,  three,  etc.,  com- 


( 


plete  vibrations  of  the  fork.  It  is  best  not  to  start  with 
the  beginning  of  the  curve,  since  the  line  may  be  more 
or  less  blurred  and  irregular  in  this  region. 

From  these  measurements,  the  acceleration  of  gravity 
can  be  determined  in  the  following  manner :  Let  VQ  be 
the  velocity  with  which  the  falling  body  passed  the  point 
<>     of  the  sinuous  curve  taken  as  origin.     Let  L  be  the  dis- 
<?     tance  from  this  point  to  an  intersection  passed  /  vibra- 
S     tions  later.    Then  we  shall  have 

CT 

(39) 

Fig.  22.  in  which  g  represents  the  acceleration  of  gravity.  If  the 
series  of  observations  taken  be  platted,  with  values  of  L  as  ordi- 
nates  and  values  of  t  as  abscissas,  the  resulting  curve  will  be  a 
parabola.  The  constants  VQ  and  g  may  be  determined  from 
this  curve  by  the  method  of  least  squares.  As  this  is  a  quad- 
ratic equation,  the  numerical  computations  will  be  very  laborious. 


MOMENT  OF  INERTIA  AND  SIMPLE  HARMONIC  MOTION.      57 

It  will  therefore  be  desirable  to  use  a  linear  equation  if  possible. 
This  may  be  done  as  follows  :  Let  /  be  the  distance  traversed 
during  the  /th  vibration  of  the  fork,  counting  from  the  assumed 
origin ;  then  we  shall  have 

/=^o+i-<r+<£*  (40) 

If  a  series  of  corresponding  values  of  /  and  t  be  platted,  this 
will  give  a  straight  line,  from  which  the  constants  v0  and  g  may 
be  determined  either  directly  by  measurement  or  indirectly  by 
the  method  of  least  squares. 

The  constants  may  also  be  derived  from  two  independent 
equations  like  the  above.  If  this  is  done,  the  two  values  of  / 
taken  should  differ  considerably,  one  being  about  twice  the  other. 

In  the  above  discussion,  the  unit  of  time  is  the  period  of  the 
fork.  Therefore  the  values  of  VQ  and  g  obtained  will  be  referred 
co  the  period  of  the  fork  as  the  unit  of  time.  Since  VQ  varies 
inversely  as  the  time,  it  is  necessary,  in  order  to  express  that 
constant  in  centimeters  per  second,  that  the  values  obtained 
be  divided  by  the  period  of  the  fork.  For  a  similar  reason,  the 
value  of  g  obtained  must  be  divided  by  the  square  of  the  fork's 
period  if  the  acceleration  of  gravity  is  to  be  expressed  in  centi- 
meters per  second  per  second. 

GROUP   E:    MOMENT    OF    INERTIA    AND    SIMPLE    HARMONIC 

MOTION. 

(E)  General  statements;  (Ex)  The  physical  pendulum;  (E2) 
Kater  s  pendulum  ;  (E3)  Relation  between  the  time  of  vibra- 
tion and  the  position  of  the  knife-edges  in  a  uniform  cylin- 
drical pendulum;  (E4)  Determination  of  the  moment  of 
inertia  of  a  body. 

(E).  General  statements  concerning  the  moment  of  inertia 
and  simple  harmonic  motion. 

The  moment  of  inertia  of  a  body,  with  respect  to  a  right  line 
taken  as  an  axis,  is  the  sum  of  the  products  of  each  element  of 


58  JUNIOR   COURSE    IN    GENERAL   PHYSICS. 

mass  by  the  square  of  its  distance  from  the  axis.  If  Ka  is  the 
moment  of  inertia  of  any  body  with  respect  to  the  axis  a,  dM 
any  element  of  mass,  and  r  its  perpendicular  distance  from  the 
axis  a,  we  have 

(40 


in  which  the  integral  is  extended  to  every  element  of  mass  of 
the  body. 

Moment  of  inertia  bears  the  same  relation  to  motion  of  rota- 
tion that  mass  bears  to  motion  of  translation.  The  following 
dynamical  relations  may  readily  be  derived  from  definitions  and 
Newton's  second  law  of  motion. 

Momentum  =  mass  x  velocity. 

Moment  of  momentum  =  moment  of  inertia  x  ang.  vel. 

Resultant  force  =  mass  x  acceleration. 

Resultant  moment  =  moment  of  inertia  x  ang.  ace. 

Kinetic  energy  —  \  mass  x  (velocity)2. 
I  Kinetic  energy  =  J  moment  of  inertia  x  (ang.  vel.)2. 

The  moment  of  inertia  of  a  body  with  respect  to  any  axis  is 
equal  to  the  moment  of  inertia  of  the  same  body  with  respect 
to  a  parallel  axis  through  the  center  of  gravity,  plus  the  mass 
of  the  body  multiplied  by  the  square  of  the  distance  between 
the  two  axes. 

Let  dM  be  an  element  of  mass  whose  co-ordinates  with 
respect  to  an  axis  through  the  center  of  gravity  are  x,  y.  Let 
c  be  an  axis  parallel  to  the  axis  through  the  center  of  gravity 
(Fig.  23)  at  distance  a  from  it.  Let  K^  and  Kc  be  the  moments 
of  inertia  with  respect  to  the  two  axes.  Then  we  shall  have 


Kc=  ft(a+x)*+f\  dM=a*CdM+(\x*+y*}dM+2  ax         or 

K.=M#+KV  (42) 

The  term    \xdM  is  zero,  for  the  origin  is  at  the  center  of 
gravity  of  the  body ;  this  means  that  the  sum  of  the  positive 


MOMENT  OF  INERTIA  AND  SIMPLE  HARMONIC  MOTION.     59 

products  xdM  is  just  equal  to  the  sum  of  the  negative  products 
-xdM. 


Fig.  23. 

Moment  of  inertia  of  an  infinitely  thin  rod  about  an  axis  per- 
pendicular to  its  length. 

Let  L  be  the  length  of  the  rod,  5  its  cross-section,  S  its 
density.  If  dx  is  the  length  of  the  element  of  mass,  we  shall 
have  dM=  SSdx.  If  x  (Fig.  24)  is  the  distance  of  the  element 


dx 


L-fc 


Fig.  24. 


of  mass  from  the  axis  a,  and  h  is  the  distance  of  the  axis  from 
either  end  of  the  rod,  we  shall  have 

(43) 


(44) 


Two  particular  cases  are  of  especial  interest ;  namely,  when 
=  O  and  when  h  =  — 


If  M  is  the  mass  of  the  whole  rod,  this  reduces  to 


60  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

Moment  of  inertia  of  a  cylinder  about  its  axis  of  figure. 

Let  L  be  the  length,  a  the  radius,  and  8  the  density  of  the 
cylinder.  Take  as  element  of  mass  an  indefinitely  thin,  hollow 
cylinder,  concentric  with  the  axis,  of  radius  r  and  thickness  dr. 

We  shall  then  have  dM=2  irrL^dr.     Therefore  we  have 

^0=2  irSL  f  V3^=J/-,  (45) 

«/o  2 

in  which  M  is  the  mass  of  the  whole  cylinder. 

Moment  of  inertia  of  a  circular  lamina  about  any  diameter. 

Let  a  be  the  radius  of  the  lamina,  e  its  thickness,  and  8  its 
density.  Taking  the  element  of  mass  in  polar  co-ordinates,  we 
have 


As  the  distance  of  the  element  of  mass  from  the  diameter  is 
?-  sin  0  (Fig.  25),  we  have 

KQ  =  2  eSj^tsin2  0a0jT  Wrj  =  M->  (46) 


Fig.  25. 

Moment  of  inertia  of  a  cylinder  about  any  axis  perpendicular 
to  its  geometrical  axis. 

Let  L  be  the  length  of  the  cylinder,  a  the  radius,  8  the 
density,  and  h  (Fig.  26)  the  distance  of  the  axis  from  one  end 
of  the  cylinder.  Taking  as  element  of  mass  a  lamina  of  thick- 
ness dx,  we  have 


MOMENT  OF  INERTIA  AND  SIMPLE  HARMONIC  MOTION.     6l 

From  equations  42  and  46  we  have  for  the  moment  of  inertia  of 
this  lamina,  with  respect  to  the  axis  a, 


dK*=x*dM+  -  dM.  (47) 

4 

*•& 

K 


/*•&-*  ,T^x>4£      /*&-* 

=  TT^B  I      x  *dx  +  ™±  I        dx, 

J-h  J-h 


and  Ka=M-hL+k*+  (48) 

L3  4j 

Two  particular  cases  are  of  especial  importance  ;  namely,  when 
/&  =  o  and  when  h  =  L. 


o 


dx 


h  \         L-7i 

Fig.  26. 

Simple  harmonic  motion. 

An  oscillating  body  is  said  to  have  simple  harmonic  motion 
when  its  distance,  either  linear  or  angular,  from  a  fixed  position 
is  a  simple  harmonic  function  of  the  time  of  either  of  the  forms, 


(49) 


in  which  A,  p,  and  tQ  are  constants. 

The  distance  of  the  body,  either  linear  or  angular,  from 
the  fixed  position  is  called  its  displacement.  The  maximum 
displacement  occurs  when  the  cosine  becomes  unity.  This 
maximum  displacement  is  called  the  amplitude  of  the  simple 
harmonic  motion.  In  the  above  equation  t§  is  a  constant  in- 
terval of  time.  This  constant  is  obviously  zero  if  time  be 
reckoned  from  the  instant  when  the  displacement  is  a  maximum 
in  the  positive  direction,  using  the  first  of  equations  49.  When 
the  time  t  has  increased  to  the  value  —  ,  the  displacement  x  is 


62 


JUNIOR   COURSE    IN    GENERAL   PHYSICS. 


exactly  equal  to  what  it  was  at  the  instant,  t—o.     Moreover,  at 
any  time,  t^  the  displacement  has  the  same  value  that  it  had  at 

The  constant  interval  of  time  — 

P 


the  time 


2  'Irn 


during  which  the  displacement  takes  all  possible  values,  and  the 
motion  begins  to  repeat  itself,  is  called  the  period  of  the  simple 
harmonic  motion.  It  is  usually  represented  by  T  or  7! 

Simple  harmonic  motion  of  translation. 

The  rectangular  projection  of  uniform  circular  motion  upon 
any  diameter  is  simple  harmonic  motion ;  i.e.  if  the  point  N 
(Fig.  27)  revolves  about  a  circle  with  a  uniform  velocity,  the 
point  P  will  move  along  the  diameter  BC  with  simple  harmonic 
motion. 

Let  A  be  the  radius  of  the  circle,  let  p  be  the  constant 
angular  velocity  of  the  radius  ON,  and  suppose  time  to  be 
reckoned  from  the  instant  that  the  point  P  leaves  the  right- 
hand  end  of  the  diameter  BC;  then 
at  any  time,  t,  we  shall  have  * 

x=A  cos  13= A  cos//.       (50) 

The    distance,   x,   is    the    displace- 
B    ment,  and  A  the  amplitude,  of   the 
simple  harmonic  motion. 

If  the  point  P  moves  through 
the  distance  dx  in  the  time  dt,  we 
have,  from  the  definition  of  velocity, 

^=-/^sin// 
dt 


Fig.  27. 


(so 


If  the  velocity  of  the  point  P  changes  by  the  amount  dv  in 
the  time  dt,  we  have,  from  the  definition  of  acceleration, 

tf=—  = 
dt 


(52). 


*  It  is  to  be  understood  that  these  equations  hold  for  any  simple  harmonic 
motion,  that  the  circle  is  an  auxiliary  circle,  and  that  the  motion  of  N  is  only  to  aid 
in  understanding  the  real  motion,  which  is  along  BC. 


MOMENT  OF  INERTIA  AND  SIMPLE  HARMONIC  MOTION.      63 

Substituting  for  A  cos//  its  value  from  equation  50,  we  have 

«=  -/*•*••  (53) 

Equation  53  shows  that  the  acceleration  of  a  point  whose 
motion  is  simple  harmonic  is  at  any  instant  proportional  to  its 
displacement  from  the '  mid-point.  The  negative  sign  shows 
that  the  acceleration  is  always  directed  oppositely  to  the  dis- 
placement ;  i.e.  when  the  point  is  at  the  right  of  the  mid-point, 
its  acceleration  is  directed  towards  the  left,  while  the  reverse 
is  true  when  the  point  is  on  the  left. 

Multiplying  both  sides  of  equation  53  by  the  mass  of  the 
moving  point,  and  remembering  the  dynamical  equation  F—Ma, 

we  have 

F=Ma=-Mp*x.  (54) 

This  is  a  dynamical  equation,  and  shows  that  the  force  which 
produces  the  acceleration  of  the  mass  M  in  simple  harmonic 
motion  is  directed  towards  the  mid-point  and  is  proportional  to 
the  displacement. 

Conversely,  it  may  be  proved  that  whenever  the  resultant 
force  acting  on  a  body  is  proportional  to  its  displacement  from 
a  fixed  position,  the  motion  of  the  body  will  be  simple  harmonic. 

From  the  equations  50,  51,  and  52,  it  follows  that  the  dis- 
placements, velocities,  and  accelerations  of  the  point  P  begin  to 
repeat  themselves  when  t  has  increased  from  o  to  -—  •  This 

?  TT 

constant  value  of  the  time is  the  period  of  the  simple  har- 
monic motion ;  and  is  obviously  the  same  as  the  time  required 
for  the  point  TV7"  to  revolve  about  the  auxiliary  circle. 

Substituting  Tior  — ,  equations  50,  51,  and  52  now  become 
P 

x=Acos^J,  (55) 

"=-sin^/,  (56) 

cost.  (57) 


64 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


It  is  often  especially  desirable  to  know  the  velocity  with 
which  the  moving  body  passes  the  mid-point.  It  will  first  pass 
the  mid-point  after  one  quarter  of  a  period  has  elapsed,  and  it 
will  pass  the  same  point  for  every  odd  number  of  quarter  periods. 
Substituting  for  t  in  the  equation  for  velocity  any  of  the  values 
\T,\T,\T>  •••,  we  have 

Simple  harmonic  motion  of  rotation. 

Let  M  be  a  body  oscillating  about  an  axis  O  (Fig.  28)  per- 
pendicular to  the  plane  of  the  paper.  The  line  OA,  fixed  in 


Fig.  28. 

the  body,  oscillates  between  the  extreme  positions  OA'  and 
OA".  The  motion  of  the  body  will  be  simple  harmonic  motion, 
according  to  the  definition  above  given,  if  we  have  at  any  time  /, 

<£  =  Scos/A  (59) 

S  and/  being  constants. 

If  d$  be  the  angle  turned  through  during  the  time  dt>  we 
shall  have,  from  the  definition  of  angular  velocity, 


<>         .*.   • 
w  =  -f-  =  —po  sin  pt. 
at 


(60) 


MOMENT  OF  INERTIA  AND  SIMPLE  HARMONIC  MOTION.      65 

If  da)  is  the  change  of  angular  velocity  in  the  time  dtt  we 
shall  have,  from  the  definition  of  angular  acceleration, 


(6  1) 

at 

Substituting  the  value  of  8  cos//  from  59,  we  have 

«=  -pt,  (62) 

from  which  it  follows  that  in  simple  harmonic  motion  of  rotation 
the  angular  acceleration  at  any  instant  is  proportional  to  the 
angular  displacement. 

Multiplying  both  sides  of  equation  62  by  the  moment  of 
inertia  of  the  rotating  body  with  respect  to  the  axis  of  rotation, 

we  have 

G=Ka=-Kp*<f>.  (63) 

Since,  however,  Ka  is  equal  to  the  resultant  moment,  with 
respect  to  the  axis  of  rotation,  of  the  forces  acting  on  the  body, 
it  follows  that  the  moment  of  the  force  producing  the  angular 
acceleration  in  simple  harmonic  motion  is  directly  proportional 
to  the  angular  displacement. 

Conversely,  it  may  be  proved  that  if  the  resultant  moment 
of  the  forces  acting  on  a  body  with  respect  to  the  axis  of  rota- 
tion is  proportional  to  the  angular  displacement  from  a  fixed 
position,  the  resulting  motion  of  the  body  will  be  a  simple 
harmonic  motion. 

Here,   as   in    simple   harmonic   motion   of   translation,   the 

motion  begins  to  repeat  itself  in  all  respects  after  a  time  — 

P 
has  elapsed.     This  constant  time  is  the  period  of   the  simple 

harmonic  motion,  and,  calling  it  T,  equations  59,  60,  and  61 
become 

=  5cos/f  (64) 


(65) 
(66) 


VOL.  I  —  F  ^^, 

Of  THE 


66  JUNIOR  COURSE   IN   GENERAL   PHYSICS. 

If  &)0  be  the  angular  velocity  with  which  the  body  passes  its 
mid-position,  we  have,  in  a  manner  similar  to  equation  58, 

"o"*^*-  (67) 

Examples  of  simple  harmonic  motion  of  translation : 

(1)  If   a   mass    be    suspended    by  a   spiral    spring,   it  will 
oscillate  along   a  vertical   line  with    simple    harmonic  motion, 
if  it  is  first  displaced  upwards  or  downwards  from  its  position 
of  equilibrium,  and  then  set  free. 

(2)  Any  molecule  in    a  sounding   body  or   a   sound-wave, 
when   the  sound   is  absolutely  simple,  i.e.  without  harmonics 
or  overtones. 

(3)  The    bob    of   a   simple    pendulum,    or   any  point   in   a 
compound  pendulum,  when  the  arc  of  vibration  is  very  small. 

(4)  Any  point  in  a  magnet  vibrating  in  a  uniform  magnetic 
field  when  the  arc  of  vibration  is  very  small. 

Examples  of  simple  harmonic  motion  of  rotation : 

(1)  A  mass  suspended  by  a  wire  or  cord,  and  rotating  about 
a  vertical  axis,  the  only  force  acting  being  the  force  of  torsion. 

(2)  A  compound  pendulum  when   the  arc  of  vibration   is 
very  small. 

(3)  A  magnet  vibrating  in  a  uniform  magnetic  field  when 
the  arc  of  vibration  is  very  small. 

In  these  examples,  as  well  as  in  all  other  cases,  there  are 
certain  retarding  forces,  as  friction,  imperfect  elasticity,  or 
induced  currents  of  electricity,  which  prevent  the  motion  from 
being  absolutely  simple  harmonic.  This  "  damping,"  as  it  is 
called,  has  an  extremely  small  effect  upon  the  period  of  the 
simple  harmonic  motion,  and  may  be  safely  neglected  when 
the  period  is  the  quantity  desired.  When  the  amplitude  of 
the  simple  harmonic  motion  is  the  quantity  to  be  used,  a 
correction  for  " damping"  must  generally  be  introduced.* 

*  See  Nichols,  The  Galvanometer,  Lecture  3;  also  Stewart  and  Gee,  Vol.  2, 
p.  364  et  seq. 


MOMENT  OF  INERTIA  AND  SIMPLE  HARMONIC  MOTION. 


67 


EXPERIMENT  Er  Determination  of  g  by  the  physical  pen- 
dulum. 

If  a  physical  pendulum  be  displaced  from  its  position  of 
equilibrium  through  an  angle  so  small  that  the  angle  may  be 
substituted  for  its  sine  without  appreciable  error,  the  moment 
of  the  force  acting  on  the  pendulum  will  be  proportional  to  the 
angular  displacement.  The  pendulum  must  therefore  have 
simple  harmonic  motion. 

From  the  principle  of  the  conservation  of  energy,  in  any 
transformation,  the  two  forms  of  energy  must  be  equal  to  each 
other.  As  the  energy  dissipated  in  a  single  swing  of  the 
pendulum  is  small  enough  to  be  negligible,  we  are  justified 
in  equating  the  kinetic  energy  of  the  pendulum  when  at  its 
lowest  point  to  the  gain  in  potential  energy  when  it  reaches 
its  highest  point. 

The  kinetic  energy  of  a  rotating  body  is  ^  Ka(o2.  Since  the 
pendulum  has  simple  harmonic  motion,  the  angular  velocity  at 
the  mid-position  will  be  -~r-  (See  equation  67.) 


The  potential  energy  at  the  highest  point  is  equal  to  the 
work  required  to  turn  the  pendulum  through  the  angle  £  from 
the  lowest  point.  This  work  is  equal  to  the  average  moment 
multiplied  by  the  angular  distance  moved,  or, 


Since  EK=EP,  we  have 


in  which  T  is  the  period  of  a  complete  oscillation,  Ka  the 
moment  of  inertia  of  the  pendulum  with  respect  to  the  axis 
of  suspension,  M  its  mass,  R  the  distance  of  the  center  of 


68 


JUNIOR   COURSE   IN    GENERAL   PHYSICS. 


gravity  from  the  axis  of  suspension,  and  g  the  acceleration 
of  gravity. 

If  T  and  R  be  observed,  and  Ka  computed,  the  acceleration 
of  gravity  may  be  determined. 

In  this  experiment  a  uniform  bar  of  metal,  provided  with 
an  adjustable  pair  of  knife-edges  (Fig.  29),  is  to  be  used  as  a 
pendulum.  The  method  of  procedure  is  as  follows  : 

(1)  Fasten  the   knife-edges  firmly 
at  some  point  not  at  the  end  of   the 
bar,  and  set  the  pendulum  to  vibrating 
through  a  small  arc. 

(2)  Determine   the   time   required 
for  the  pendulum  to  make  some  large 
number  of  oscillations. 

(3)  From  the  result  compute   the 
period  of  the  pendulum.     An  ordinary 
watch  or  clock  may  be  used  for  deter- 
mining   this    time,   although    a   stop- 
watch is  better.      In  any  case,  several 

determinations  of  the  period  should  be  made,  in  each  of  which 
the  time  is  at  least  five  or  six  minutes. 

(4)  After  the  period  has  been  determined,  measure  the 
dimensions  of  the  bar  and  the  distance  of  the  knife-edges  from 
one  end.  From  these  data,  the  moment  of  inertia  can  be 
computed. 

As  the  bar  is  homogeneous,  the  center  of  gravity  will  be  at 
the  center  of  the  figure,  and  thus  R  is  known.  M  will-  be  found 
to  cancel  out ;  consequently  g  may  be  computed. 

The  knife-edges  and  clamp  slightly  affect  the  moment  of 
inertia  and  the  center  of  gravity  of  the  pendulum,  thus  slightly 
changing  the  period.  If  greater  accuracy  is  desired,  the  effect 
of  the  knife-edges  and  clamp  on  the  period  may  be  made  zero 
by  fastening  to  the  knife-edges  an  auxiliary  mass,  a  portion  of 
which  extends  above  the  axis  of  suspension,  and  varying  the 
center  of  gravity  of  this  mass  until  the  period  of  vibration  of 


Fig.  29. 


MOMENT  OF  INERTIA  AND  SIMPLE  HARMONIC  MOTION.     69 


the  knife-edges  without  the  bar  is  approximately  the  same  as 
the  period  of  the  bar  pendulum. 

The  observations  and  results  of  the  experiment  are  to  be 
tabulated  in  the  manner  indicated  below  : 


GRAVITY  BY  THE  PHYSICAL  PENDULUM. 


No.  of  Transit 
to  Right. 

Time. 

Duration  of  100 
Oscillations. 

Dimensions  of  Pendulum  and  Results. 

hr.    min.    sec. 

Distance  of  axis  from  upper 

I 

27    oo 

end  of  bar,                                 3.5  cm. 

IOI 

30      24 

204S. 

Length  of  bar,                      L=  159.6  cm. 

201 

33    49 

205 

Diameter  of  bar,                 zr—     1.6  cm. 

3OI 

37     13 

204 

Distance  of  axis  from  center 

4OI 

40    38  £ 

205  1 

of  gravity  of  bar,            R=  76.3  cm. 

hr.    min.    sec. 

Moment  of  inertia,            Ka  —  7945  M. 
Periodic  time,                       7^=2.047 

I 

42      50 

Computed  value  of  gravity, 

IOI 

46      14 

204 

g—  981+  cm.  per  sec.  per  sec. 

201 

49    40 

2O6 

Most  careful   determination 

3OI 

53      3 

203 

for  Cornell  laboratory,             980.28 

4OI 

56    29 

206 

Addenda  to  the  report: 

(1)  Explain  how  the  mass  of  the  pendulum  cancels  so  that 
it  does  not  need  to  be  known.     Compute  the  acceleration  of 
gravity  in  centimeters  per  second  per  second  and  in  feet  per 
second  per  second. 

(2)  Explain  what  is  meant  by  the  statement,  The  force  of 
gravity  is  g  dynes. 

(3)  Explain  why  a  small  error  in  the  period  will  make  an 
error  relatively  twice  as  great  in  g. 

EXPERIMENT  E2.     Determination  of  g  by  Kater's  pendulum. 

The  equation  for  the  physical  pendulum  may  be  put  into  the 
form 

(69) 


jO  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

in  which  JT0  is  the  moment  of  inertia  of  the  pendulum  with 
respect  to  an  axis  through  the  center  of  gravity  parallel  to  the 
axis  of  suspension.  (See  equations  42  and  68.)  If  the  pendu- 
lum be  inverted,  and  the  time  of  vibration  determined  for  an 
axis  on  the  opposite  side  of  the  center  of  gravity,  a  new 
equation  will  be  obtained  similar  to  the  above,  except  that  there 
will  be  new  values  of  R  and  T.  Between  these  two  equations 
KQ  may  be  eliminated  and  g  determined.  To  this  end  we 
substitute  /  for  Rl  +  R2  in  the  equation  derived  by  the  elimina- 
tion of  KQ  from  two  equations  like  the  above.  It  will  then 
reduce  to  the  equation  for  the  simple  pendulum,  provided  that 
the  two  times  of  oscillation  are  the  same,  and  the  two  values  of 
R  are  not  the  same.  Kater's  pendulum  is  an  apparatus  which 
makes  use  of  this  fact. 

The  use  of  Kater's  pendulum  depends  upon  the  principle 
that  the  center  of  oscillation  and  the  center  of  suspension  of 
any  pendulum  are  interchangeable ;  i.e.  if  a  pendulum  is  reversed, 
so  that  the  point  which  was  the  center  of  oscillation  is  made  the 
center  of  suspension,  the  time  of  vibration  will  remain  unchanged. 
The  distance  between  these  two  points  being  equal  to  the  length 
of  the  corresponding  simple  pendulum,  the  measurement  of  this 
length,  together  with  the  observation  of  the  time  of  vibration, 
is  sufficient  to  determine  the  force  of  gravity.  The  experiment 
consists,  therefore,  in  adjusting  the  positions  of  the  two  knife- 
edges  by  trial  until  the  time  of  vibration  about  one  pair  as  an 
axis  is  the  same  as  that  about  the  other. 

The  pendulum  used  consists  of  a  hollow  cylindrical  bar, 
one  end  of  which  is  loaded  by  a  filling  of  lead  (Fig.  30).* 
There  are  two  pairs  of  knife-edges,  one  being  placed  near 
each  end  of  the  bar  ;  both  are  capable  of  adjustment  along  the 
bar,  so  that  the  distance  between  them  can  be  altered. 

*  The  pendulum  here  referred  to  is  a  very  simple  one,  but  with  careful  observa- 
tions is  capable  of  giving  quite  accurate  results.  A  homogeneous  cylindrical  bar  is 
sometimes  used,  but  with  such  a  pendulum  one  pair  of  knife-edges  will  be  at  such  a 
point  that  considerable  variation  of  its  position  will  produce  but  little  change  in  the 
time  of  vibration.  (See  Exp.  E3,  and  Fig.  31.) 


MOMENT  OF  INERTIA  AND  SIMPLE  HARMONIC  MOTION. 


The  method  of  the  experiment  is  as  follows  : 

(1)  Fasten  one  pair  of  knife-edges  to  the  bar  at  some  point 
near  the  end  which  is  not  weighted. 

(2)  Determine  the  rate  of  vibration  roughly  by  observing 
with  a  watch  or  clock  the  time  occupied 

by  some   large   number  of   oscillations 
(20-50). 

(3)  Locate   approximately  the  posi- 
tion of  the  center  of  oscillation  by  hang- 
ing a  simple  pendulum  (a  small  weight 
suspended    by    a    cord)    near   by,    and 
adjusting   its   length    until    it   vibrates 
nearly   in    unison   with    the   bar.     The 
length  of  this  simple  pendulum  is  then 
a  rough  approximation  to  the  distance 
from    the  center  of   suspension   of   the 
bar   to    its    center   of    oscillation.      To 
obtain   this   distance    more    accurately, 
set  the  second  pair  of  knife-edges  at  a 
distance    from    the   first    equal   to   the 

length  just  determined ;  then  reverse  the  bar,  and  determine  its 
time  of  vibration  as  before.  The  period  should  now  be  nearly 
the  same  as  at  first.  If  the  two  periods  differ,  one  or  both  of 
the  knife-edges  should  be  shifted  until  the  time  of  vibration 
is  very  closely  the  same  with  either  suspension. 

(4)  The   final   determination  of  the  time  of  vibration  must 
be  made  very  carefully,  and  should  be  repeated  several  times. 

(5)  Finally,  the  distance  between  the  knife-edges  is  to  be 
measured   as  accurately  as  possible.       From  this  distance  and 
the  two  times  of  vibration,  the  value  of  g  is  to  be  computed. 

Determine  the  value  of  g  in  the  C.  G.  S.  system,  and  also  in 
the  foot-pound-second  system. 


Fig.  30. 


JUNIOR   COURSE    IN    GENERAL   PHYSICS. 


EXPERIMENT  E3.  Relation  between  the  time  of  vibration  and 
the  position  of  the  knife-edges  in  a  uniform  cylindrical  pendulum. 

In  the  equation  for  the  physical  pendulum  given  in  the 
preceding  experiment,  everything  is  determined  for  any  given 
pendulum  except  R  and  T.  The  object  of  this  experiment  is  to 
show  the  relation  which  exists  between  these  two  variables. 

To  this  end,  fasten  the  knife-edges  at  one  end  of  the  bar, 
and  determine  the  period  of  vibration  by  observing  the  time 
occupied  by  a  large  number  of  oscillations.  Then  shift  the 


3.6 


3.2 


O  2.8 

a 
2 

£T 


2.0 


1.6 


1O  2O  3O  4O  50  6O  7O  SO  90 

DISTANCE  OF  KNIFE-EDGE  FROM  CENTER  OF  PENDULUM 
Fig.  31. 

knife-edges  down  the  bar  three  or  four  centimeters,  and  deter- 
mine the  new  period.  Continue  shifting  the  point  of  suspension 
and  observing  the  period  until  the  center  of  the  bar  is  reached. 
From  eight  to  ten  different  positions  of  the  knife-edges  should 
be  used,  and  the  distance  of  the  knife-edges  from  one  end  of 
the  bar  should  be  carefully  measured  in  each  case.  In  deter- 
mining the  time  of  vibration,  a  stop-watch  will  be  found  of 
considerable  assistance.  From  the  data  obtained,  plat  a  curve, 
similar  to  that  given  in  Fig.  31,  using  for  abscissas  the  distances 


MOMENT  OF  INERTIA  AND  SIMPLE  HARMONIC  MOTION. 


73 


of  the  point  of  suspension  from  the  center  of  the  bar,  and  for 
ordinates  the  corresponding  times  of  vibration.  Discuss  and 
explain  the  shape  of  this  curve,  and  determine  the  form  of  its 
equation  from  a  knowledge  of  the  law  of  the  physical  pendulum. 
The  following  is  a  tabulated  statement  of  a  set  of  observations 
taken  as  indicated  above.  The  results  are  shown  graphically  in 
Fig.  31. 

RELATION  OF  PERIODIC  TIME  TO  POSITION  OF  KNIFE-EDGES. 


Distance 
from  Center 
of  Gravity. 

No.  of 
Transit  to 
Right. 

Time. 

Duration  of 

IOO 

Oscillations. 

Periodic 
Time. 

Other  Data  and  Results. 

hr.  min.  sec. 

I 

2    16   50 

Length  of  bar  =183  cm. 

90 

IOI 

2O    30 

2  2O 

2.21 

Diam.     "     "    =2.5    " 

201 
I 

24    12 
2Q    OO 

222 

Equation  of  curve  from 

80 

IOI 

J7 

32   34 

214 

2.15 

theory  of  pendulum  : 

201 

36    10 

216 

2           4  W2£2 

I 

42  oo 

T^R  =  -  —  R^1  +  —  — 

70 

IOI 

45    30 

210 

2.10 

g                 S 

201 
I 

49  oo 

3    22    OO 

2IO 

From  pairs  of  points  on 

60 

IOI 

25     27 

2O7 

2.07 

curve   having   equal  ordi- 

2O  I 

28   54 

207 

nates. 

55 

I 
IOI 

38  25 

205 

2.O5 

T=2.20,       £-=970; 

20  1 

4i   5° 

205 

T—  2.16,       £-=989; 

5° 

IOI 

51   4° 
55     6 

206 

2.O6 

T=2.i2,       <r=977- 

20  1 

58  32 

2O6 

I 

4     6   30 

40 

IOI 

10       I 

211 

2.10 

201 

13   30 

209 

I 

17    10 

3° 

IOI 

20    52^ 

222^ 

2.225 

201 

24    35" 

2221 

I 

28    40 

20 

IOI 

32   54 

254 

2-535 

2O  I 

37     7 

253 

I                     42     40 

10 

ioi               48   24 

344 

3435 

201               54     7 

343 

Addenda  to  the  report: 

(i)  The  equation  of  this  curve  expressed  in  its  simplest 
form  will  contain  two  constants.  Show  what  relation  these  con- 
stants bear  to  the  pendulum  and  to  the  acceleration  of  gravity. 


74  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

(2)  From  any  pair  of   points    on  the    curve    compute  the 
values  of  these  constants,  and  hence  determine  the  acceleration 
of  gravity. 

(3)  Prove  that  the  abscissa  corresponding  to  the  minimum 
ordinate  is  equal  to  the  radius  of  gyration  of  the  pendulum  with 
respect  to  its  center  of  gravity. 

EXPERIMENT  E4.  Determination  of  the  moment  of  inertia 
of  a  body. 

The  pendulum  furnishes  a  means  of  determining  the  moment 
of  inertia  with  respect  to  an  axis  through  the  center  of  gravity 
of  any  body  to  which  a  pair  of  knife-edges  may  be  attached, 
and  whose  center  of  gravity  may  be  determined. 

In  the  equation  for  the  physical  pendulum  given  in  experi- 
ment E2,  K0  may  be  computed  if  Tt  M,  and  R  be  determined, 
g  being  known. 

Take  any  body  whose  mass  is  great  with  respect  to  the  knife- 
edges  to  be  used ;  fasten  the  knife-edges  to  it  at  some  distance 
from  the  center  of  gravity.  Determine  the  period  of  oscillation 
as  in  any  of  the  preceding  experiments ;  weigh  the  body,  and 
measure  the  distance  of  the  knife-edges  from  the  center  of 
gravity.  From  the  data  thus  obtained  compute  the  moment 
of  inertia,  KQ.  Repeat  the  same  observations  and  computations 
for  two  or  three  other  bodies. 

If  any  of  the  bodies  are  regular  solids,  check  the  values 
thus  obtained  by  direct  integration,  as  described  at  the  begin- 
ning of  this  chapter.  Express  the  results  obtained  both  in  the 
C.  G.  S.  system  and  in  the  foot-pound-second  system. 

GROUP  F:  ELASTICITY. 

(FJ)    Young's  modulus ;  (F2)  Moment  of  torsion;  (F3)  Moment 
of  inertia  by  torsion. 

EXPERIMENT  Fv     Young's  modulus  by  stretching. 
Elasticity  of  tension  is  denned  as  the  ratio  of  force  applied 
to  extension  produced.     If  the  elastic  limit  has  not  been  reached, 


ELASTICITY. 


the  extension  of  a  weighted  rod  or  wire  is  proportional  (i)  to 
the  force  applied,  (2)  to  the  length,  (3)  inversely  proportional 
to  the  cross-section,  or 


in  which  E  is  the  coefficient  of  elasticity.  From  this  equation 
it  is  seen  that  E  is  the  increase  in  length  produced  by  unit 
force  applied  to  a  rod  of  unit  length  and  unit  cross-section. 

Young's  modulus  is  denned  as  the  reciprocal  of  the  coeffi- 
cient of  elasticity.     Calling  this  modulus  M,  we  have 


<"> 

Young's  modulus  may  be  computed  if  the  quantities  on  the 
right  of  this  equation  are  determined  in  the  proper  units. 

Fasten  a  wire  two  or  three  meters  in  length  to  some  firm 
support.  A  small  vice  rigidly  attached  to  the  brick  or  stone 
wall  of  the  room  makes  a  support  which  is  satisfactory  in  most 
cases.*  Suspend  from  the  end  of  the  wire  a  weight  which  is 
just  sufficient  to  take  out  the  kinks  ;  for  a  wire  whose  diameter 
is  i  mm.  a  weight  of  from  two  to  four  kilos  will  be  required. 

A  horizontal  microscope  containing,  an  eye-piece  microm- 
eter is  now  to  be  adjusted  so  that  a  slight  scratch  on  the 
wire  is  sharply  focused  in  the  lower  part  of  the  field.  As  the 
tension  of  the  wire  is  increased  by  the  addition  of  weights, 
this  mark  will  move  across  the  field,  and  by  means  of  the 
micrometer  the  elongation  corresponding  to  each  increment 
in  weight  can  be  measured.  Measure  in  this  way  the  elonga- 
tions produced  by  successive  increments  in  weight  until  the 
mark  has  passed  out  of  the  field.  Each  increment  in  weight 
should  be  sufficient  to  cause  an  elongation  of  three  or  four  scale 
divisions. 

*  If  there  is  any  reason  to  suspect  that  the  support  is  not  rigid,  two  microscopes 
must  be  used,  one  at  the  upper  and  the  other  at  the  lower  end  of  the  wire. 


76  JUNIOR   COURSE    IN    GENERAL   PHYSICS. 

After  the  wire  has  been  fully  loaded  the  weight  is  to  be 
gradually  reduced  and  the  measurements  repeated  until  the 
original  load  is  reached.  Note  whether  equal  increments  of 
tension  produce  equal  increments  of  length,  and  whether  the 
elastic  limit  has  been  passed. 

The  results  can  be  best  shown  by  a  curve  in  which  abscissas 
represent  force  applied,  and  ordinates  the  increments  in  length 
produced. 

Determine  the  value  of  one  division  of  the  micrometer  as 
described  in  Exp.  A4  (III),  and  measure  the  length  and  diameter 
of  the  wire. 

Since  the  square  of  the  radius  enters  in  the  above  equation, 
a  small  error  in  determining  it  will  be  relatively  doubled  in 
the  computed  value  of  the  modulus.  For  this  reason  the 
diameter  of  the  wire  must  be  measured  with  unusual  care. 
The  cross-section  may  be  most  accurately  determined  by  com- 
putation from  the  density  and  the  mass  of  a  given  length. 
Since  the  density  of  various  specimens  is  liable  to  differ, 
the  density  should  be  carefully  determined  by  weighing  in 
water. 

EXPERIMENT  F2.  Determination  of  the  moment  of  torsion 
of  a  wire. 

When  a  wire  of  elastic  material,  such  as  steel,  bronze,  or 
hard  drawn  copper,  is  twisted  by  a  moderate  amount,  the 
moment  of  the  couple  by  which  it  tends  to  regain  its  original 
condition  is  proportional  to  the  angle  of  torsion ;  i.e.  if  Q  is  the 
angle,  and  G  the  moment  of  the  elastic  return  force,  G=GQ6. 
The  constant  GQ  is  called  the  moment  of  torsion,  and  depends 
upon  the  length,  diameter,  and  material  of  the  wire. 

To  determine  the  value  of  GQ,  a  heavy  weight,  of  such  shape 
that  its  moment  of  inertia  can  be  readily  computed,  is  hung 
upon  the  end  of  the  wire,  and  set  to  vibrating  through  an  angle 
of  twenty  or  thirty  degrees.  Since  the  moment  of  the  return 
force  is  proportional  to  the  angular  displacement,  the  weight 


ELASTICITY.  jj 

will  have  simple  harmonic  motion,  and  the  vibration  will  be 
isochronous.  From  equation  63  we  will  have 

(72) 

U,l,~ 

in  which  G06  is  the  resultant  moment  due  to  torsional  displace- 

/2/3 

ment  through  an  angle  0,  and  — -is  the  angular  acceleration  of 
the  suspended  weight.  An  integration  of  this  equation  gives 

(73) 

in  which  T  is  the  period  of  the  harmonic  motion. 

The  same  equation  may  be  derived  more  easily  from  the 
energy  relations.  If  8  is  the  maximum  angular  displacement, 
the  kinetic  energy  of  the  rotating  weight  as  it  passes  the  mid- 
position  will  be 

The  potential  energy  of  the  twisted  wire,  when  the  suspended 
weight  is  at  its  greatest  displacement,  is  equal  to  the  work  that 
must  be  done  on  the  wire  to  twist  the  lower  end  through  the 
angle  8.  The  moment  of  the  force  at  any  instant  to  be  over- 
come is  GQ0  ;  as  this  varies  between  o  and  GQ8,  the  average 
moment  is  J  G08,  and  hence  the  work  done  and  the  potential 
energy  gained  is 

£P=±GQ8.8.  (75) 

As  the  dissipation  of  energy  during  a  single  vibration  may  be 
neglected,  the  potential  energy  at  the  extreme,  when  the  weight 
has  no  motion,  must  be  equal  to  the  kinetic  energy  when  the 
weight  is  at  its  mid-position  and  there  is  no  twist  in  the  wire. 
Hence  we  have 

r=27rVJ-  <76) 

To  determine  the  period  of  vibration,  the  method  of  Exp.  A5 
should  be  used.  It  is  to  be  observed  that  since  it  is  the  square 


;8  JUNIOR   COURSE   IN    GENERAL   PHYSICS. 

of  T  that  appears  in  the  formula,  an  error  in  the  determination 
of  the  period  will  introduce  a  considerable  error  in  the  result. 
The  moment  of  inertia  is  to  be  computed  from  the  mass  and 
linear  dimensions  of  the  vibrating  weight. 

From  the  result  obtained  for  G0  compute  the  force,  both  in 
dynes  and  in  pounds,  which  would  twist  the  wire  through  a 
complete  revolution  when  acting  at  a  distance  of  one  centimeter 
from  the  center. 

EXPERIMENT  F3.  To  determine  the  moment  of  inertia  of  an 
irregular  body  by  torsion. 

The  preceding  experiment  offers  one  of  the  best  means  of 
determining  the  moment  of  inertia  of  an  irregular  body.  GQ  is 
a  constant  for  any  given  wire,  independent  of  the  mass  sus- 
pended by  it  and  of  the  period  of  oscillation.  Therefore,  if  it  is 
already  known,  the  moment  of  inertia  of  the  suspended  weight 
may  be  computed  after  the  period  has  been  determined.  If  GQ 
is  not  known,  it  may  be  eliminated  between  two  equations  of 
the  form  (76),  in  one  of  which  the  moment  of  inertia  of  the 
suspended  weight  is  known. 

To  perform  the  experiment,  suspend  the  body  by  a  wire  of 
phosphor  bronze  or  some  other  elastic  material,  the  upper  end 
of  the  wire  being  rigidly  fastened.  The  axis  about  which  the 
moment  of  inertia  is  required  should  lie  in  the  prolongation 
of  the  wire.  Set  the  system  to  vibrating,  and  determine  the 
period  as  in  the  previous  experiment.  Then  hang  upon  the 
wire  a  body  whose  moment  of  inertia  is  known,  and  determine 
the  vibration  period  as  before.  If  the  two  periods  are  Tv  T2, 
then 

*2  =  *i  (77) 


CHAPTER   II. 
GROUP  G:  DENSITY. 

(G)  General  statements ;  (G1)  Rough  determination  of  specific 
gravity  by  weighing  in  water ;  (G2)  Specific  gravity  of  solids 
and  liquids  by  the  specific  gravity  bottle  ;  (G3)  Determination 
of  density  with  corrections  for  air  displacement  and  tem- 
perature;  (G4)  Specific  gravity  by  the  Jolly  balance; 
(G5)  Nicholson  s  hydrometer ;  (G6)  Fahrenheit 's  hydrometer ; 
(G7)  Graduation  of  a  hydrometer  of  variable  immersion; 
(G8)  Specific  gravity  of  a  solid  by  means  of  a  variable  immer- 
sion hydrometer  ;  (G9)  Density  of  a  liquid  by  Hare's  method. 

(G.)  General  statements  concerning  specific  gravity  and 
density. 

The  specific  gravity  of  a  substance  is  the  ratio  of  the  weight 
of  a  given  volume  of  the  substance  to  the  weight  of  an  equal 
volume  of  water  at  its  maximum  density. 

Specific  gravities  being  ratios  of  like  quantities  are  abstract 
numbers,  and  hence  the  same  for  all  systems  of  units.  The  unit 
used  in  comparing  the  weights  may  indeed  be  entirely  arbitrary, 
such  as  the  unit  extension  of  a  spring  made  use  of  in  the  Jolly 
balance. 

The  density  of  a  substance  is  the  ratio  of  the  mass  of  a 
given  volume  of  the  substance  to  the  volume  which  it  occupies ; 
or  in  symbols 

/>=£•  (78) 

Since  density  is  not  an  abstract  number,  its  numerical  value 
in  any  particular  case  must  depend  upon  the  units  used.  For 

79 


8o  JUNIOR   COURSE   IN    GENERAL   PHYSICS. 

example,  the  density  of  water  in  the  foot-pound-second  system 
is  62|.  The  density  of  water  in  the  C.  G.  S.  system  is  unity, 
for  the  reason  that  the  unit  of  mass  is  equal  to  the  mass  of  a 
cubic  centimeter  of  water.  Hence  it  follows  that  the  densities 
of  all  substances  in  the  C.  G.  S  system  are  numerically  equal 
to  their  specific  gravities.  This  is  not  absolutely  true,  however, 
for  the  mass  of  a  cubic  centimeter  of  water  at  its  maximum 
density  is  not  exactly  a  gram. 

The  term  "  relative  density  "  is  sometimes  used.  It  has  the 
same  meaning  as  "specific  gravity." 

EXPERIMENT  Gr  Rough  determination  of  specific  gravity 
by  weighing  in  water. 

Specific  gravity  of  a  body  more  dense  than  water.  —  Weigh 
the  body  in  air;  then  suspend  it  from  a  hook  under  one  of  the 
scale-pans  of  a  balance,  immerse  it  in  water,  and  weigh  again. 
The  specific  gravity  is  to  be  computed  from  these  two  weights, 
no  correction  being  made  for  the  temperature  of  the  water  or 
the  buoyancy  of  the  air.  The  wire  used  for  suspending  the 
body  must  be  quite  fine.  It  should  be  immersed  in  water  to 
the  same  extent  that  it  will  be  when  the  body  is  attached,  and 
balanced  with  shot  or  sand,  before  the  second  weighing  above 

is  made. 

II. 

Specific  gravity  of  a  body  lighter  than  water.  —  First  weigh 
the  body  in  air.  Then  suspend  a  heavy  sinker  from  one  scale- 
pan,  and  find  its  weight  when  immersed  in  water.  Finally 
attach  the  body  to  the  sinker,  and  find  the  weight  of  the  two 
when  entirely  submerged.  From  these  three  weights  compute 
the  specific  gravity. 

The  results  should  be  tested  by  placing  the  vessel  of  water 
on  the  scale-pan  and  suspending  the  substance  in  the  water  from 
some  outside  support.  The  gain  in  weight  of  the  vessel  should 
be  the  same  as  the  loss  of  the  substance. 


DENSITY.  8l 

Errors  in  this  method  of  determining  specific  gravity  are  apt 
to  arise  from  small  bubbles  of  air  adhering  to  the  substance 
when  immersed ;  such  bubbles  must,  therefore,  be  carefully 
shaken  off  before  the  weighings  are  made. 

EXPERIMENT  G2.  Specific  gravity  of  solids  and  liquids  by 
the  specific  gravity  bottle. 

The  specific  gravity  bottle  is  simply  a  small  bottle  which  is 
provided  with  an  accurately  fitting  ground-glass  stopper.  A 
very^mall  hole  through  the  center  of  this  stopper  leads  to  the 
interior  of  the  bottle,  its  object  being  to  allow  the  bottle  to  be 
completely  filled  with  any  liquid. 

To  use  the  specific  gravity  bottle,  proceed  as  follows : 

I. 

Specific  gravity  of  a  liquid.  —  First  weigh  the  bottle  alone, 
when  perfectly  clean  and  dry.  Next  fill  with  distilled  water  and 
weigh  again.  Finally  fill  the  bottle  with  the  liquid  whose 
density  is  required,  and  weigh  a  third  time.  These  three 
weights  are  sufficient  for  the  computation  of  the  specific 
gravity. 

II. 

Specific  gravity  of  a  solidl  —  Place  the  substance  in  the 
specific  gravity  bottle  and  determine  the  combined  weight. 
Then  add  sufficient  distilled  water  to  entirely  fill  the  bottle, 
insert  the  glass  stopper,  and  after  wiping  off  any  drops  which 
may  adhere  to  the  outside,  weigh  again.  Finally  determine  the 
weight  of  the  bottle  when  filled  with  water  alone.  These  three 
weights,  together  with  the  weight  of  the  bottle,  are  sufficient  to 
determine  the  specific  gravity  of  the  substance.  This  method 
is  of  course  only  available  when  the  substance  is  insoluble  in 
water.  In  the  case  of  soluble  substances  some  liquid  of  known 
density  must  be  used  in  which  the  substance  does  not  dissolve. 


1  The  specific  gravity  bottle  is  especially  useful  when  the  solid  is  in  the  form  of 
small  fragments  or  powder. 

VOL.  I  —  G 


82  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

It  sometimes  happens  that  difficulty  is  met  with  in  shaking 
off  the  small  bubbles  of  air  which  tend  to  adhere  to  the  sub- 
stance, and  which  will  introduce  a  considerable  error.  In  such 
cases  the  bottle  containing  the  substance,  and  about  half  full  of 
water,  should  be  placed  under  the  receiver  of  an  air-pump,  and 
the  air  exhausted  until  bubbles  are  no  longer  formed. 

If  greater  accuracy  is  required,  corrections  for  temperature 
and  air  displacement  must  be  made,  similar  to  those  described 
in  Exp.  G3. 

EXPERIMENT  G3.  Determination  of  density,  with  corrections 
for  air  displacement  and  temperature. 

The  balance  is  nearly  always  used  for  comparing  masses, 
but  it  should  be  remembered  that  it  is  merely  a  lever  with  equal 
arms,  by  which  two  forces  may  be  proved  to  be  equal.  Each  of 
the  two  equal  forces  is  the  resultant  of  the  weight  of  the  body 
on  the  scale-pan  acting  downwards,  and  the  buoyant  effect  of 
the  weight  of  the  fluid  displaced  by  the  body  acting  upwards. 

This  gives 

W^-w^  *F2-w/2. 

Since  weights  are  directly  proportional  to  masses,  we  have 

M1-m1  =  M2-wzt  (79) 

in  which  M1  and  M2  are  the  masses  of  the  bodies  on  the  two 
scale-pans,  and  m^  and  m^  are  the  masses  of  the  displaced  fluid 
in  the  two  cases.  Nearly  always  in  using  the  balance,  m1  and 
m2  are  supposed  to  be  equal,  or  at  least  it  is  assumed  that  their 
difference  is  negligible.  ^ 

If  M,  is  the  mass  of  the  substance  of  density  Ss,  then  from 
the  definition  of  density  the  volume  of  the  displaced  fluid  will 

be  — .     If  Ba  is  the  density  of  the  displaced  air,  then  its  mass 

S, 
<> 

is  M,—.    If  M  is  the  mass  of  the  counterpoise,  and  Sc  its  density, 

S« 
equation  79  becomes 

(So) 


DENSITY.  83 

In  order  to  determine  Ms  from  the  known  mass  of  the  counter- 
poise, Sat  &•>  and  Sc  must  be  known.  Approximate  values  for 
these  quantities  will  serve  quite  as  well  as  more  accurate  values, 
because  the  term  in  which  they  appear  is  always  a  very  small 
quantity. 

The  object  of  this  experiment  is  to  determine  the  density  of 
the  substance  s  with  all  possible  accuracy.  If  the  substance  is 
suspended  from  the  scale-beam,  so  as  to  be  immersed  in  water 
of  density  p,  and  is  then  counterpoised  with  the  mass  Mf,  equa- 
tion 79  becomes 

M,  —  Ma—=M'  —  M'^.  (81) 

If  equation  80  be  divided  by  81,  and  the  resulting  equation  solved 
for  Ss,  we  shall  have 

Ss  =  — ^—^(p-Sa).  (82) 

In  deriving  equation  82  it  has  been  assumed 

(1)  That  the  density  of  the  counterpoise  M  is  the  same  as 
that  of  M'. 

(2)  That  the  density  of  the  air  has  not  changed  between  the 
two  weighings. 

(3)  That  the  density  of  the  substance  or  of  the  weights  has 
not  been  changed  during  the  experiment  on  account  of  expan- 
sion.    The  value  of  p  depends  on  the  temperature  of  the  water. 
The  value   of   Sa  depends  on    the   temperature,   pressure,   and 
humidity  of  the   atmosphere  at    the   time    of   performing   the 
experiment.     The  effect  of  humidity  in  altering  the  density  of 
the  air  may  be  neglected  except  when  the  substance  weighed  is 
a  gas  or  a  vapor. 

Use  the  most  accurate  balances  that  are  available,  counter- 
poise the  substance  whose  specific  gravity  is  required,  first  in 
air,  and  then  when  suspended  by  a  fine  wire,  in  distilled  water. 
Observe  also  the  temperature  of  the  water  and  the  temperature 
and  barometric  pressure  of  the  atmosphere.  The  distilled  water 


84 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


used  should  first  be  thoroughly  boiled  in  order  to  expel  the 
dissolved  air. 

The  values  of  p  for  different  temperatures  can  be  found  in 
most  reference  books,  while  Sa  can  be  computed  from  the  tem- 
perature and  pressure  of  the  air.* 

In  this  experiment  all  observations  must  be  taken  with  great 
care.  A  suitable  correction  should  be  made  for  the  weight  of 
the  wire  used  in  suspending  the  substance  in  water,  and  all  air 
bubbles  that  may  adhere  to  the  wire  or  specimen  must  be  care- 
fully removed. 

When  the  value  of  8S  is  finally  obtained,  it  must  be  remem- 
bered that  this  is  the  density  of  the  substance  at  the  tempera- 
ture of  the  water  in  which  it  was  weighed.  For 
comparison,  this  must  be  reduced  to  o°  by  using 
the  coefficient  of  cubic  expansion. 

EXPERIMENT  G4.  Specific  gravity  by  the  Jolly 
balance. 

The  Jolly  balance  consists  of  a  spiral  spring 
hanging  in  front  of  a  vertical  graduated  scale,  and 
carrying  at  its  lower  end  two  small  scale-pans. 
The  lower  of  these  should  always  be  kept  immersed 
in  water,,  as  shown  in  Fig.  32,  and  in  order  to 
render  this  possible,  the  bracket  which  supports 
the  vessel  of  water  is  made  adjustable  in  position. 

The  instrument  may  be  used  in  determining 
specific  gravity  in  two  different  ways : 

I. 

Fig  32  Place  the  body  whose  specific  gravity  is  required 

upon  the  upper  scale-pan  and  observe  the  elonga- 
tion of  the  spring.  The  weight  of  the  body  is  now  determined 
by  finding  what  known  weight  is  required  to  produce  the  same 


*  Tabulated  values  of  p  and  5a  will  be  found  in  Landoldt  and  Bornstein,  in 
Stewart  and  Gee,  Vol.  I,  etc. 


DENSITY. 


elongation.  Then  place  the  body  on  the  lower  scale-pan  (under 
water),  and  observe  what  weight  must  be  placed  on  the  upper 
pan  to  make  the  elongation  the  same  as  before.  These  two 
observations  are  evidently  sufficient  to  determine  the  specific 
gravity. 

II. 

The  specific  gravity  may  also  be  determined  without  the 
use  of  weights,  upon  the  assumption  that  the  elongation  of 
the  spring  is  proportional  to  the  force  tending  to  stretch  it. 

The  specific  gravity  of  some  solid  should  be  determined 
by  each  of  the  above  methods,  and  the  assumption  made  in 
method  II  (i.e.  that  the  elongation  is  proportional  to  the 
weight)  should  be  tested  by  observing  the  elongations  pro- 
duced by  five  or  six  different  weights.  Compute  the  value  in 
grams  of  an  elongation  of  one  scale  division.  Find  also  the 
number  of  dynes  required  to  produce  an  elongation  of  one 
division. 


EXPERIMENT  G5. 


Specific  gravity  by  Nicholson's  hydrom- 


eter. 


This  hydrometer   (Fig.   33)   consists    of   a   hollow   cylinder 
which  is  made  to  float  with  its  rifk 

axis  vertical  by  means  of  a  heavy 
weight  at  the  bottom.  At  the 
top  a  wire  projects  two  or  three 
inches  above  the  end  of  the 
cylinder  and  supports  a  small 
scale-pan.  At  the  bottom  an- 
.  other  pan  is  provided,  upon 
which  can  be  placed  the  object 
whose  density  is  required. 

To   determine    the    specific  _~E  

gravity   of    a    solid,    place    the  Fis-  33. 

hydrometer  in  water,  and  find  by  trial  the  weight  which  must 
be  placed  on  the  scale-pan  in  order  to  bring  some  well-defined 


86  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

mark  to  the  surface  of  the  water.  In  the  instrument  shown  in 
Fig.  33>  which  is  a  slight  modification  of  the  hydrometer 
of  Nicholson,  this  mark  consists  of  the  point  of  a  wire  which 
projects  downward  from  the  center  of  the  scale-pan.  Then 
place  upon  the  scale-pan  the  body  whose  density  is  required, 
and  add  weights  until  the  instrument  has  sunk  again  to  the 
same  level.  Finally  place  the  body  upon  the  lower  pan  or 
basket,  and  again  determine  the  weight  necessary  to  sink  the 
hydrometer.  From  these  three  weights  the  specific  gravity 
can  be  computed.  In  case  the  specimen  is  lighter  than  water 
it  must  be  fastened  in  some  way  to  the  bottom  of  the  instrument 
to  prevent  it  from  floating  away.  The  instrument  may  also 
be  used  in  determining  the  specific  gravity  of  a  liquid. 

This  form  of  hydrometer  is  not  very  sensitive,  and  there- 
fore cannot  be  expected  to  give  results  of  great  accuracy.  In 
this  experiment,  however,  as  in  all  specific  gravity  determi- 
nations, the  most  common  source  of  error  is  the  presence 
of  air  bubbles,  which  will  adhere  both  to  the  specimen  and 
to  the  instrument  unless  carefully  shaken  off. 

The  report  should  contain  a  full  explanation  of  the  principles 
involved,  including  Archimedes'  Law. 

EXPERIMENT  G6.     Fahrenheit's  hydrometer. 

This  hydrometer  consists  of  an  elongated  glass  bulb 
(Fig.  34),  weighted  at  the  bottom,  and  carrying  at  the  top  a 
small  scale-pan  supported  by  a  wire  sealed  into  the  bulb. 

To  determine  the  density  of  a  liquid,  first  float  the  instrument 
in  distilled  water  and  place  weights  on  the  scale-pan  until  some 
well-defined  mark  on  the  stem  is  brought  to  the  surface  of  the 
water.  It  will  be  found  preferable  to  use  bits  of  tin-foil  for 
weights.  The  tin-foil  corresponding  to  each  separate  obser- 
vation should  then  be  wrapped  in  a  piece  of  paper,  labeled,  and 
afterwards  weighed  on  a  pair  of  balances.  Then  place  the 
hydrometer  in  the  liquid  whose  specific  gravity  is  required  and 
determine  the  weight  necessary  to  sink  it  to  the  same  point. 


DENSITY. 


From  these  two  weights,  together  with  the  weight  of  the 
hydrometer,  the  specific  gravity  of  the  liquid  can  be  computed. 
A  correction  should  be  made  for  the  temperature  of  the  water. 

I 

Use  the  instrument  in  the  manner  just  described,  to  deter- 
mine the  variation  in  the  density  of  a  salt  solution  as  its  degree 
of  concentration  is  altered.  To 
accomplish  this,  first  dissolve  in 
water  sufficient  salt  to  make  a 
nearly  saturated  solution,  weigh- 
ing both  the  salt  and  the  water. 
Having  determined  the  density 
of  this  solution,  dilute  it  by  the 
addition  of  a  known  weight  of 
water,  and  again  determine  its 
density.  Continue  in  this  way 
until  the  solution  is  so  dilute  as 
to  have  nearly  the  same  specific 
gravity  as  water.  At  least  eight 
or  ten  different  observations  Fi8f>  34- 

should  be  taken.  With  the  results  obtained,  plat  a  curve  in 
which  the  strengths  of  the  solution  are  used  as  abscissas  and 
the  corresponding  densities  as  ordinates. 

II. 

Fahrenheit's  hydrometer  may  be  used  for  determining  the 
density  of  water  at  different  temperatures.  In  this  experi- 
ment it  will  not  be  allowable  to  assume  that  the  volume  of  the 
submerged  portion  of  the  hydrometer  is  constant,  for  as  the 
temperature  is  changed,  the  hydrometer  expands  or  contracts. 
The  volume  of  the  hydrometer  at  temperature  /  may  always  be 
expressed  in  the  form  F0(i +/£/),  in  which  F0  is  its  volume  at 
o°,  and  k  is  the  coefficient  for  cubical  expansion  of  glass. 

If  M  is  the  mass  of  the  hydrometer,  m  the  additional  mass 


88  JUNIOR   COURSE   IN    GENERAL   PHYSICS. 

necessary  to  sink  the  hydrometer  to  the  point  of  reference,  and 
St  the  density  of  the  water  at  the  temperature  t,  we  shall  have 


(83) 

If  another  observation  be  now  taken  in  which  t,  and  conse- 
quently m,  are  different,  we  shall  have  another  equation  similar 
to  the  above.  Between  these  two  equations  F0  may  be  elim- 
inated, and  the  ratio  of  Bt  to  St-  determined.  If  one  of  these 
values  is  already  known,  the  other  may  be  computed  in  absolute 
measure. 

For  this  experiment  fill  a  vessel  nearly  full  of  distilled  water, 
and  cool  it  by  means  of  ice  and  salt  down  to  2°  or  3°  C.  Make 
eight  or  ten  determinations  of  corresponding  values  of  m  and  t. 
The  values  of  /  should  differ  by  approximately  equal  increments, 
with  two  or  three  additional  observations  at  temperatures  as 
near  as  possible  to  4°. 

From  the  observations  near  4°,  determine  by  interpolation 
the  value  of  m  that  would  correspond  to  a  temperature 
of  4°.  Assume  the  density  at  4°  to  be  unity,  and  compute 
the  densities  at  the  other  observed  temperatures.  From  the 
results  thus  obtained,  plat  a  curve  with  temperatures  as  abscissas 
and  densities  as  ordinates.  The  ordinates  should  be  on  a  greatly 
enlarged  scale,  the  axis  of  abscissas  not  being  shown  at  all. 
Upon  the  same  paper,  plat  the  results  of  some  standard  deter- 
mination. 

EXPERIMENT  G7.  Graduation  of  a  hydrometer  of  variable 
immersion. 

The  density  of  liquids  is  very  frequently  determined  by 
means  of  variable  immersion  hydrometers.  These  hydrometers 
consist  of  an  elongated  glass  bulb  weighted  at  the  bottom  with 
mercury,  and  supporting  a  graduated  stem  of  uniform  cross- 
section.  The  graduations  on  the  stem  are  not  equidistant,  how- 
ever, for  equal  increments  of  submersion  in  different  liquids  do 
not  correspond  to  equal  decrements  in  density. 


DENSITY. 


89 


If  J/is  the  mass  of  the  hydrometer,  V§  the  volume  of  that 
part  below  the  lowest  division  of  the  scale,  a  the  cross-section 
of  the  stem,  and  /  the  added  length  of  stem  submerged  in  a 
liquid  of  density  8,  we  have,  from  Archimedes'  principle  and 
the  definition  of  density, 

M=(VQ  +  ta)8  =  (V0  +  /fa)S' '  =  (F04-/"tf)S" '  =  •••.          (84) 

The  product  of  density  by  volume  submerged  is  a  constant. 
Therefore  the  volumes  submerged  in  different  liquids  vary 
inversely  as  their  densities.  It  also  follows  that  as  the 
densities  increase  in  arithmetical  progression,  the  vol-  a, 
umes  submerged  must  decrease  in  a  corresponding  har-  az 
monic  progression.  For  example,  if  the  series  of  densi- 
ties is  i,  i.i,  1.2,  1.3,  •••,  the  series  of  volumes  must  be  5 

numerically  ^,  ^,  ^,  ^~,     If  M,  F0,  and  a  were 
i     i.i     1.2    1.3 

known,  the  values  of  /  could  be  computed  corresponding 
to  any  particular  arithmetical  series  of  densities.  How- 
ever, if  the  values  of  two  points  on  the  scale  be  experi- 
mentally determined  as  described  below,  it  will  be  A 
possible  to  determine  the  values  of  other  points  by  the 
harmonic  law. 

Suppose  ^  and  #5,  Fig.  35,  to  be  the  two  experi-  Figr' 3J 
mentally  determined  points  of  the  scale  corresponding  to  den- 
sities of  i  and  1.4,  respectively.  Let  it  be  required  to  find  a 
point  a2  which  shall  correspond  to  a  density  of  i.i.  Let  A  be 
a  point  such  that  distances  measured  from  A  to  av  a2,  •••,  will  be 
proportional  to  the  volume  of  the  hydrometer  up  to  these 
points.  Then  we  shall  have 

If  Aa&  be  eliminated  between  these  equations,  the  position 
of  #2  may  be  determined,  that  of  av  a5  being  already  known. 
Other  points  may  be  determined  in  the  same  way. 

The  graduation  may  be  performed  much  more  easily  and 
quite  as  accurately  by  the  following  graphic  method.  Let  it  be 


90  JUNIOR   COURSE    IN    GENERAL   PHYSICS. 

remembered  that  the  end  will  be  attained  if  the  scale  is  divided 
harmonically  so  that  the  fixed  points  a±  and  a5  shall  correspond 
to  densities  I  and  1.4,  respectively.  Take  any  line  PPl  (Fig.  36), 
and  divide  it  so  that  PPlt  PPZ,  •••,  PP*>  are  m  the  harmonic 

progression  i,  — ,  — ,  — ,  — .     From  any  point,  O,  at  a  con- 
i.i     1.2    1.3    1.4 

venient  distance  from  PP^  draw  lines  through  the  points  Plt 
Pz,  •••.  Any  line  drawn  across  this  series  of  diverging  lines 


Fig.  36. 

parallel  to  PP^  will  be  divided  by  them  in  the  same  harmonic 
progression  as  PP\.  Now  take  the  paper  scale  and  lay  it 
across  these  lines  (keeping  it  parallel  to  />/>1)  so  that  a^  will  fall 
on  the  line  drawn  through  Plt  and  a5  will  fall  on  the  line  drawn 
through  P6.  If  the  distances  between  the  graduations  so  deter- 
mined are  not  great,  these  distances  may  be  subdivided  into 
equal  parts  without  introducing  an  appreciable  error. 

To  determine  experimentally  the  two  fixed  points,  proceed 
as  follows : 

Place  the  instrument  in  distilled  water,  and  adjust  the  paper 
scale  in  the  tube  until  its  zero  is  at  the  level  of  the  water.  Next 
determine  (by  one  of  the  methods  of  Exp.  G2  or  G9)  the  density 
of  some  liquid  considerably  heavier  than  water  (e.g.  a  strong 


DENSITY.  91 

salt  solution).  Place  the  hydrometer  in  this  liquid  and  observe 
the  reading.  Having  now  two  points  on  the  scale,  the  inter- 
mediate divisions  can  be  determined  by  either  of  the  methods 
described  above. 

Finally  test  the  accuracy  of  the  calibration  by  using  the 
hydrometer  to  measure  the  specific  gravities  of  one  or  two  liquids 
of  intermediate  density. 

EXPERIMENT  G8.  Density  of  a  solid  by  means  of  a  variable 
immersion  hydrometer.* 

If  a  hydrometer  of  variable  immersion  is  provided  with  two 
pans,  one  above  and  the  other  below  the  surface  of  the 
liquid,  it  may  be  used  for  the  determination  of  the  density 
of  a  solid. 

Let  a  be  the  cross-section  of  the  stem,  /  the  added  length  of 
the  stem  submerged  when  a  substance  of  mass  m  is  placed 
on  the  upper  scale-pan.  From  Archimedes'  principle,  the 
increased  mass  of  liquid  displaced  must  equal  the  mass  of  the 
substance.  Therefore,  we  have,  from  the  definition  of  density, 

m  =  a®,  (86) 

in  which  8  is  the  density  of  the  liquid  in  which  the  hydrometer 
is  placed.  If  the  substance  is  placed  in  the  lower  pan,  the  same 
volume  of  liquid  will  be  displaced  as  before,  but  since  the  sub- 
stance itself  is  below  the  surface,  a  shorter  length  of  the  stem 
will  be  submerged.  And  we  have 

^-8,  (86  a) 


~  being  the  volume  of  the  mass  m.    From  (86)  and  (86  a)  we  have 

68 

(87) 


8     /-/' 

From  this  equation  it  follows  that  either  8e  or  8  may  be  deter- 
mined if  the  other  is  known. 

*  This  form  of  hydrometer  is  due  to  G.  H.  Failyer  of  the  Kansas  Agricultural 
College. 


92 


JUNIOR   COURSE   IN    GENERAL   PHYSICS. 


To  perform  the  experiment,  first  observe  the  division  of  the 
scale  which  coincides  with  the  surface  of  the  liquid  when  no 
substance  is  placed  on  the  hydrometer.  Next  observe  the 
scale  reading  when  the  substance  is  placed  successively  on 
the  upper  and  lower  scale-pans. 

From  these  observations  the  density  of  the  subtance  may  be 
computed  if  the  density  of  the  liquid  is  known. 

Determine  in  this  way  the  densities  of  several  solids.  The 
determination  will  be  most  accurate  when  the  sample  tested 
is  as  large  as  possible. 

To  verify  the  statement  that  the  volume  of  liquid  displaced 
by  a  floating  body  is  proportional  to  its  mass,  proceed  as  follows : 
Place  a  known  mass  on  the  upper  pan,  and  observe  the  corre- 
sponding  scale   reading.      Repeat   these   observations   with   a 
series  of  masses,  varying  from  zero  to  the  maxi- 
mum   that    can  be   used.      Plat   a   curve  with 
masses  as  abscissas  and  scale  readings  as  ordi- 
nates.     Show  that  this  curve  verifies  the  above 
statement.     From  its  constants    determine  the 
cross-section  of  the  stem,  assuming  the  density 
of  the  liquid  to  be  unity. 

EXPERIMENT  G9.     Density  of  a  liquid  by 
Hare's  method. 

The  apparatus  used  in  this  experiment  con- 
sists of  two  vertical  tubes  open  below  and  con- 
nected above  to  a  common  tube ;  the  latter 
tube  is  provided  with  stop-cock  (see  Fig.  37). 
The  two  tubes  dip  into  separate  vessels,  one 
containing  distilled  water,  and  the  other  the 
liquid  whose  density  is  to  be  determined.  The 
p.  37  tubes  are  fastened  to  an  upright  board  on  which 

there  is  a  scale. 

If  the  pressure  of  the  air  in  the  common  tube  is  reduced 
by  suction,  the  liquid  will  rise  in  each  tube,  the  heights  of 


PROPERTIES   OF   GASES. 


93 


the  two  columns  being  inversely  proportional  to  the  densities 
of  the  liquids  used. 

This  may  be  demonstrated  as  follows  :  Let  a  be  the  atmos- 
pheric pressure,  b  the  pressure  of  the  air  in  the  common  tube 
above  the  two  columns  of  liquid,  both  measured  in  dynes  per 
square  centimeter.  Let  k,  h'  ,  and  S,  &'  be  the  heights  and 
densities  of  the  two  columns  of  liquid.  From  Pascal's  law  we 
have  for  any  point  within  the  first  tube  on  a  level  with  the 
surface  of  the  liquid  in  the  open  vessel, 

a  =  b  +  hfa  (88) 

and  for  the  corresponding  point  within  the  second  tube, 


Put  distilled  water  into  one  of  the  vessels,  and  the  liquid 
whose  density  is  to  be  determined  into  the  other.  By  suction 
cause  the  liquids  to  rise  in  the  tubes  until  the  top  of  the  highest 
column  is  near  the  end  of  the  scale.  Adjust  the  level  of  the 
liquid  in  each  vessel  until  it  is  at  the  zero  of  the  scale,  and 
read  the  heights  of  the  two  columns.  Then  open  the  stop- 
cock until  the  columns  have  fallen  through  6  or  8  cm.  Adjust 
as  before,  and  again  read  the  height  of  each  column.  Repeat 
these  readings  for  several  different  heights. 

Compare  in  this  way  the  densities  of  three  different  liquids 
with  that  of  distilled  water.  The  tubes  should  be  rinsed  with 
distilled  water  before  and  after  using  each  different  liquid. 

GROUP  H:   PROPERTIES  OF  GASES. 

(Hj)  Verification  of  Boyle  s  law  ;  (H2)  Comparison  of  the  cistern 
barometer  and  the  siphon  barometer;  (H3)  Coefficient  of 
expansion  of  air. 

EXPERIMENT  Hr     Verification  of  Boyle's  law. 

The  apparatus  consists  of  two  glass  tubes  mounted  vertically 
upon  some  suitable  support  and  connected  at  the  bottom.  One 


94 


JUNIOR   COURSE    IN   GENERAL   PHYSICS. 


tube  is  left  open  at  the  top,  while  the  other  can  be  closed  so  as 
to  be  air  tight.  Both  are  provided  with  scales  to  enable  the 
height  of  the  mercury  contained  in  them  to  be  measured.  (See 
F'g.  38.) 

To  test  the  law  for  pressures  greater  than  one  atmosphere. — 
For  this  purpose  one  tube  should  be  considerably  shorter  than 
the  other. 

(i)  In  case  the  two  tubes  are  not  provided  with  a  common 
scale,  determine  two  points,  one  on  each  scale,  which  are  at 
the  same  level.  This  can  be  done  by  observing 
the  height  to  which  mercury  rises  in  the  two 
tubes  when  both  are  open  to  the  air ;  or  the 
same  thing  may  be  accomplished  by  means  of  a 
spirit  level. 

(2)  The  end  of  the  shorter  tube  being  tightly 
closed,  observe  the  height  of  the  mercury  in 
each  tube.  Then  increase  the  pressure  by  pour- 
ing more  mercury  into  the  longer  tube,  and 
again  observe  the  two  levels.  Continue  in  this 
way  until  the  longer  tube  is  filled  nearly  to  the 
top,  taking  in  all  about  ten  observations.  To 
check  these  observations  the  pressure  should 
now  be  gradually  diminished  by  allowing  mer- 
cury to  escape  from  the  stop-cock  at  the  bottom 
of  the  apparatus.  Ten  more  readings  should  be 
taken  as  the  pressure  falls  to  its  original  value. 
In  each  of  the  observations  above,  the  total  pressure  to 
which  the  air  in  the  short  tube  is  subjected  is  measured  by  the 
difference  in  level  between  the  two  columns  of  mercury  phis 
the  pressure  of  the  atmosphere.  In  tabulating  the  results  each 
difference  in  level  should  therefore  be  increased  by  the  height 
of  the  barometer  at  the  time  of  the  experiment. 

If  the  tube  containing  the  air  is  of  uniform  cross-section,  the 
volume  of  the  confined  air  is  proportional  to  the  length  of  the 


Fig.  38. 


PROPERTIES    OF   GASES.  95 

tube.  In  this  experiment  it  is  sufficiently  accurate  to  assume 
the  tube  to  be  uniform,  except  at  the  closed  end,  where  the  cross- 
section  is  apt  to  be  irregular.  The  zero  point  of  the  scale  used 
with  the  shorter  tube  is  therefore  placed,  not  at  the  top  of  the 
tube,  but  a  little  below  the  top.  If  /  is  the  reading  on  this 
scale,  and  VQ  the  unknown  volume  of  that  portion  of  the  tube 
above  the  zero  point,  then  the  total  volume  is  V=  V^  +  IA,  in 
which  A  is  the  cross-section.  If  Boyle's  law  is  true,  we  should 
havePF=^;  or  P(V^  +  IA)  =  K.  With  the  exception  of  P  and 
/,  all  the  quantities  in  this  equation  are  constant.  If  a  curve  is 
platted  with  the  observed  values  of  /  as  abscissas  and  the  cor- 
responding values  of  i-t-P  for  ordinates,  this  curve  should 
therefore  be  a  straight  line.  Determine  the  equation  of  this 
line  by  the  method  of  least  squares,  and  from  this  equation 
compute  the  values  of  V§  and  K.  Reduce  both  quantities  to 
C.  G.  S.  units. 

The  cross-section  A  may  be  determined  by  weighing  a  small 
amount  of  mercury  which  is  allowed  to  escape  from  the  appa- 
ratus, and  at  the  same  time  observing  the  alteration  in  the 
readings  of  the  two  columns.  Since  the  density  of  mercury  is 
known,  these  observations  are  sufficient  for  the  computation 
of  A. 

If  more  accurate  results  are  desired,  the  short  tube  must  be 
calibrated  by  means  of  mercury,*  and  the  air  used  must  be  care- 
fully dried.  In  all  cases  great  care  must  be  taken  to  keep  the 
temperature  of  the  air  constant. 

The  student  will  find  it  interesting  to  compute  the  constant 

PV 

k  in  the  equation  =£,  which  is  true  for  a  perfect  gas  at  all 

0 

temperatures.  If  this  is  done,  the  results  should  be  put  in  such 
a  form  as  to  refer  to  the  volume  and  pressure  of  one  gram  of 
air.  It  will  then  be  possible  to  compare  the  value  computed  for 
k  with  those  given  in  various  reference  books,  and  a  check  on 
the  results  of  the  whole  experiment  is  obtained. 

*  See  Stewart  and  Gee,  vol.  I. 


96 


JUNIOR  COURSE   IN    GENERAL   PHYSICS. 


II. 

To  test  the  law  for  pressures  less  than  one  atmosphere.  —  In 
this  case  the  two  tubes  should  be  of  the  same  length.  Both 
tubes  are  first  filled  with  mercury  to  within  10-20  cm.  of  the 
top.  One  tube  is  then  tightly  closed,  and  observations  are 
taken  as  the  pressure  is  reduced  by  drawing  off  mercury  from 
the  bottom.  The  pressure  should  then  be  gradually  increased 
again  until  the  confined  air  has  returned  to  its  original  condition. 

In  other  respects  this  experiment  is  exactly  the  same  as  the 
preceding. 

EXPERIMENT  H2.  Comparison  of  a  cistern  barometer  and 
a  siphon  barometer. 

The  apparatus  for  this  experiment  consists  of  two  barom- 
eters of  the  types  indicated   above,  an  accurate  vertical  scale 
A  a     divided  to  millimeters,  and  a  read- 

ing telescope  mounted  upon  a  ver- 
tical rod.  The  length  of  this  rod 
should  be  at  least  80  cm.,  and  the 
vertical  scale  should  be  of  the 
same  length.  It  is  essential  that 
the  telescope  turn  freely  upon  its 
support  with  an  accurately  hori- 
zontal motion. 

The  arrangement  of  the  appa- 
ratus which  is  shown  in  Fig.  39  is 
as  follows : 

The  two  barometers  are  mount- 
ed side  by  side  upon  a  substantial 
block.  At  points  a  and  b,  situated 
at  distances  equally  distant  to  the 
right  of  barometer  B+  and  to  the 


B, 


Fig.  39. 


left  of  barometer  B^  are  pins  from  which  the  scale  5  may  be 
suspended.     The   latter    must  be  adjusted  beforehand  so  that 


PROPERTIES   OF   GASES.  97 

when  the  A -shaped  opening  is  placed  on  either  pin  the  scale 
will  swing  freely  into  a  vertical  position. 

The  reading  telescope  should  be  set  up  at  as  small  a  distance 
from  the  barometers  as  the  length  of  the  draw-tube  will  permit, 
and  should  be  in  such  a  position  that  the  meniscus  of  either 
mercury  column  can  be  seen,  and  also  the  scale,  in  good  defini- 
tion, without  change  of  focus. 

These  adjustments  having  been  completed,  the  following 
observations  are  to  be  made  : 

(1)  Scale  hanging  at  the  right. 

(a)  The  telescope  is  focused  upon   the   upper  meniscus  of 
barometer  Bl  (siphon),  and  the  distance  from  the  cap  of  the 
meniscus  to  the  fixed  cross-hair  in  the  eye-piece  is  measured 
by  means  of  a  micrometer.* 

(b)  The  telescope   is   then    swung   to  the   right   until   the 
vertical  scale  comes  into  the  field.     (In  case  the  scale  is  not 
in  proper  focus,  further  adjustment  must  be  made  by  moving 
it  towards  or  away  from  the  telescope,  and  not  by  refocusing 
the  latter.) 

(c}  The  scale  divisions  nearest  the  fixed  cross-hair  are  identi- 
fied and  noted,  and  their  distances  from  the  latter  are  measured 
by  means  of  the  micrometer  screw. 

(d)  These  operations  are  repeated  in  the  case  of  barometer 
B<i  (cistern). 

(2)  Scale  hanging  at  the  left. 

(e)  The  various  operations  described  as  a,  b,  c,  and  d  are 
carefully  repeated. 

(/)  The  telescope  is  shifted  to  a  position  opposite  the  cis- 
tern of  barometer  B^  and  the  level  of  the  mercury  in  the  same 
is  obtained  by  readings  similar  to  those  described  under  a,  b, 
and  c. 

*  In  case  the  reading  telescope  is  not  provided  with  a  micrometer  eye-piece,  the 
common  eye-piece  should  be  furnished  with  a  suitable  ruling  on  glass,  which,  placed 
in  the  focus,  makes  a  very  good  substitute. 

""-'-"  *&& 

THS 


98  JUNIOR   COURSE    IN   GENERAL   PHYSICS. 

(g)  The  level  of  the  lower  meniscus  of  barometer  B^  is 
determined  as  above. 

(3)  Scate  hanging  at  the  right. 

(h)  The  levels  of  cistern  and  lower  meniscus  are  redeter- 
mined  as  above. 

(4)  The   reading  of  a  thermometer  placed  midway  between 
the  two  mercury  columns  is  noted. 

If  the  conditions  indicated  in  the  description  of  this  experi- 
ment are  fulfilled,  that  is  to  say,  if  the  scale  hangs  vertically 
both  at  the  right  and  left,  and  the  telescope  moves  smoothly  in 
a  nearly  horizontal  plane,  the  height  of  mercury  column  (B1  and 
B^  respectively)  will  be  found  nearly  the  same,  whether  com- 
puted from  readings  with  scale  left  or  scale  right.  Any  dis- 
crepancy approaching  o.oi  cm.  should  indicate  the  advisability  of 
repeating  the  measurements.  The  height  of  the  two  mercury 
columns  in  Bl  and  B^  will,  however,  differ  very  appreciably,  even 
when  the  vacuum  is  good  in  both  instruments.  The  difference 
is  due  to  depression  by  capillary  action,  which  influences  the 
cistern  barometer  only.  The  next  step  is  to  determine  whether 
the  correction  for  capillarity  will  account  for  the  difference  of 
barometric  height. 

(5)  To  calibrate  the  cistern  barometer  for  capillarity,  note  the 
reading  of  the  meniscus  when  the  screw  by  means  of  which  the 
height  of  the  mercury  in  the  cistern  *  is  adjusted,  is  at  almost 
its  lowest  position ;   then  add  a  weighed  quantity  of  pure  mer- 
cury to  the  cistern  sufficient  to  produce  a  rise  of  about  one  cen- 
timeter in  the  surface  of  the  contents.     The  meniscus  will  rise 
through  a  distance  precisely  corresponding  to  the  change  of  level 
in  the  cistern,  and  in  case  the  ratio  in  the  cross-sections  be  not 
very  large  indeed,  the  change  of  level  as  compared  with  that  which 

*  The  cistern  barometer  to  be  used  in  this  experiment  should  be  provided  with  a 
cistern  which  has  a  flexible  leather  bottom,  upon  which  a  screw  impinges  as  in  the 
Fortin  barometer,  giving  considerable  range  of  level. 


PROPERTIES   OF   GASES.  99 

would  have  occurred  had  there  been  no  loss  of  mercury  from  the 
cistern  to  supply  the  increase  in  the  column  within  the  baro- 
metric tube,  will  afford  a  fair  approximation  to  the  diameter  of 
the  latter.  This  determination  involves  the  measurement  of  the 
dimensions  of  the  cistern  and  the  computation  of  its  contents 
per  centimeter  of  vertical  height. 

In  case  the  difference  in  the  observed  height  for  Bl  and  B^ 
is  not  entirely  accounted  for  by  means  of  the  correction  for 
capillarity  (concerning  which  see  any  one  of  the  larger  treatises 
in  physics),  it  is  probable  that  the  vacuum  in  one  or  both 
barometers  is  imperfect.  Gross  errors  of  filling  may  be  detected 
by  driving  the  column  of  Bz  to  the  top  of  the  tube,  by  means 
of  the  screw,  and  watching  for  a  bubble  which  cannot  be  made 
to  disappear  by  pressure,  and,  in  the  case  of  the  siphon  barom- 
eter, reaching  the  same  end  by  the  direct  application  of  pressure 
to  the  open  end  of  the  tube. 

To  reduce  the  readings  obtained  in  this  experiment  to 
absolute  measure,  the  scale  should  be  placed  upon  the  dividing 
engine  (Exp.  A3),  and  compared  with  some  good  standard  of 
length,  or  with  the  screw  itself,  if  the  constant  of  the  instru- 
ment is  known. 

EXPERIMENT  H3.     The  coefficient  of  expansion  of  air. 

The  apparatus  used  in  this  experiment  is  a  modification  of 
that  of  Regnault.  It  consists  of  an  air  thermometer  bulb, 
B  (Fig.  40),  contained  in  a  jacketed  vessel  which  serves  as 
a  bath  of  constant  temperature.  This  bulb  is  connected  with 
a  mercury  manometer  by  means  of  a  long  metallic  tube  of 
very  small  bore. 

The  coefficient  of  expansion  is  to  be  indirectly  determined 
from  the  changes  of  pressure  necessary  to  maintain  the  air 
contained  within  the  bulb  at  a  constant  volume  when  subjected 
to  changes  of  temperature. 

Unless  the  bulb  has  been  previously  filled  with  dry  air, 
it  must  be  so  filled  by  connecting  it  to  an  air-pump,  and 


100 


JUNIOR  COURSE    IN    GENERAL   PHYSICS. 


exhausting  several  times.  After  each  evacuation,  air  is  allowed 
to  enter  the  bulb  through  the  set  of  intervening  drying  tubes 
(71,  Fig.  40).  The  "three-way"  stopcock  J^is  arranged  to  admit 
of  this  operation  without  the  necessity  of  detaching  the  air- 
bulb  from  the  manometer. 

After  having  been  thus  pumped  out  and  refilled  at   least 
ten   times,  the   bulb   is   to   be   brought    into    connection  with 


Fig.   40. 

the  manometer  by  turning  the  stopcock  V.  There  should  be 
no  further  communication  between  its  contents  and  the 
exterior  atmosphere.  The  two  temperatures  for  which 
measurements  are  to  be  made  are  that  of  melting  ice  and 
that  of  boiling  water.  To  obtain  the  former,  the  bulb  is 
entirely  surrounded  by  crushed  ice,  and  is  kept  thus  packed 
until  all  upward  movement  of  the  mercury  column  in  the 
closed  arm  of  the  manometer  ceases.  To  prevent  an  over- 


PROPERTIES    OF   GASES.  IOI 

flow  of  the  mercury  into  the  neck  of  the  bulb,  the  pressure 
must  be  repeatedly  lowered  by  adjustment  of  the  iron  plunger 
P,  by  small  amounts,  to  compensate  for  the  tendency  of  the 
gas  to  contract.  When  the  temperature  of  the  bulb  has  become 
constant,  the  pressure  is  raised  until  the  mercury  reaches  the 
mark  a  in  the  neck  of  the  manometer.  The  height  of  both 
mercury  columns  is  then  observed,  preferably  by  the  aid  of  a 
reading  telescope  and  auxiliary  scale,  as  described  in  Exp.  H2, 
and  the  height  of  the  barometer  is  determined. 

After  the  completion  of  these  readings,  the  ice  is  removed 
from  the  constant  temperature  bath,  a  proper  amount  of  hot 
water  is  supplied,  the  Bunsen  burner  is  lighted,  and  the  bulb 
is  subjected  to  the  continued  action  of  steam  until  it  reaches 
the  temperature  corresponding  to  that  which  steam  assumes  at 
the  pressure  under  which  it  is  generated  within  the  bath. 
During  the  process  of  heating,  the  pressure  upon  the  air  within 
the  bulb  should  be  readjusted  from  moment  to  moment,  so  that 
no  considerable  deviation  of  the  mercury  from  the  mark  a 
takes  place.  Finally,  after  equilibrium  at  the  temperature  of 
steam  has  been  reached,  a  careful  readjustment  of  the  mercury 
column  to  the  mark  is  made,  and  this  operation  is  followed  by 
readings  of  the  manometric  pressure  and  of  the  barometer. 

In  addition  to  these  data  it  is  necessary  to  know  the 
contents  of  the  bulb  at  o°  C.,  and  at  the  temperature  of  the 
steam  bath ;  also  to  apply  certain  corrections. 

For  the  purposes  of  practice  work,  it  is  well  to  have  a  value 
of  the  cubic  contents  of  the  bulb  previously  determined  with 
care  once  for  all.  In  this  way,  by  assuming  the  accuracy  of 
this  value  and  making  use  of  the  same  in  computation,  the 
tedious  process  of  refilling  with  dry  air  before  each  repetition 
of  the  experiment  may  be  avoided. 

The  process  of  determining  the  contents  may  be  learned, 
and  the  measurement  of  the  coefficient  of  expansion  of  the 
glass  may  be  performed  by  using  a  duplicate  bulb.  This  bulb 
is  filled  with  mercury  at  o°,  and  the  contents  is  weighed, 


102 


JUNIOR   COURSE    IN  GENERAL   PHYSICS. 


for  which  purpose  it  is  divided  into  as  many  parts  as  the 
capacity  of  the  balance  may  make  necessary.  It  is  then  filled 
at  the  temperature  of  steam,  and  the  contents,  after  cooling, 
is  weighed.  The  mean  coefficient  k  is  expressed  as  follows  : 

£=(~J-i)J>  (89) 

where  t  is  the  temperature  of  the  steam  bath.  The  tempera- 
ture t  is  to  be  determined  from  the  pressure  of  the  vapor  in 


80. 


76. 


74. 


72. 


70. 


98 


99' 


Fig.  41.  — Boiling  Point  and  Pressure. 

the  steam  bath,  for  which  purpose  the  results  of  Regnault  may 
be  used.  That  portion  of  his  curve  which  applies  to  pressures 
in  the  neighborhood  of  one  atmosphere  is  given  in  Fig.  41.* 
By  means  of  it  the  temperature  of  a  steam  bath  for  any  ordinary 
barometric  pressure  can  be  obtained  without  the  use  of  a 
thermometer. 

To  compute  Vol0  and  Vol,,  the  density  of  mercury  must  be 
known  at  o°  and  at  t°.  This  may  be  conveniently  obtained 
from  the  curve  of  densities,  given  in  Fig.  42.* 


*  For  the  data  from  which  the  curves  are  obtained,  see  Landolt  and  Bornstein, 
Tabellen,  pp.  41  and  58. 


PROPERTIES   OF   GASES. 


103 


The  most  important  corrections  are  those  arising  from  the 
depression  of  the  mercury  in  the  neck  of  the  manometer  at  a, 
and  from  the  temperature  of  the  mercury  in  the  manometric 
and  barometric  columns.  The  former  is  to  be  ascertained  by 
isolating  the  bulb  from  the  manometer,  by  turning  V  to 


13.58 
13.56 
13.54 
13.52 
13.50 
13.48 
13.46 
13.44 
13.42 
13.4O 
13.38 
13.36 

\ 

N 

\ 

J 

NSIT 

YOI 

ME 

ICUF 

Y 

\ 

\ 

\ 

\ 

\ 

f\ 

s 

S 

\ 

\ 

\ 

\ 

> 

0° 


20' 


4O  °  6O 

Fig.  42. 


100 


the  proper  position,  and  bringing  the  mercury  to  the  mark 
by  means  of  the  plunger.  The  difference  of  level  at  a  and 
in  the  open  tube  when  the  two  surfaces  are  subjected  to  the 
same  (the  barometric)  pressure  is  then  noted.  This  reading  is 
most  accurately  performed  by  means  of  the  vertically  suspended 
scale  already  described,  and  the  reading  telescope.  The  latter 
correction  is  that  which  it  is  necessary  to  apply  to  all  barometric 
and  manometric  readings  on  account  of  the  change  of  density 
of  the  mercury  with  temperature,  viz. 


*»=*.£• 

Un 


(90) 


where  hn  is  the  corrected  barometric   or  manometric   height, 


104  JUNIOR   COURSE    IN    GENERAL   PHYSICS. 

//,  the  observed  height,  and  —  is  the  ratio  of  the  densities  of 

"o 
mercury  at  the  temperature  of  observation  and  at  o°  C. 

A  less  important  correction  than  the  foregoing  is  that  due 
to  tha  fact  that  a  certain  volume  of  the  air  contained  in  the 
apparatus,  that,  namely,  in  the  tube  connecting  the  bulb  with 
the  neck  of  the  manometer  does  not  change  temperature.  If 
the  bulb  is  of  considerable  size  (300  cu.  cm.  or  more),  and  the 
tube  has  the  contracted  bore  described  above,  the  volume  under 
consideration  will  be  found  to  be  of  little  influence.  The 
student  should,  however,  make  the  measurements  and  compu- 
tation necessary  to  assure  himself  of  the  fact.  For  this  purpose 
a  piece  of  the  tubing  which  is  used  in  the  apparatus  is  provided. 
The  diameter  of  this  is  to  be  gauged  and  the  contents  of  the 
connecting  tube  estimated  therefrom,  and  from  the  length  of 
the  latter. 

The  neck  of  the  manometer  from  a  to  the  joint  b  is 
frequently  of  greater  volume  than  the  connecting  tube.  Its 
contents  may  be  directly  determined  as  follows  : 

(1)  By  means  of  the  plunger  drive  the  mercury  into  the 
neck  of  the  manometer  until  it  reaches  the  stopcock  V,  which 
must  have  been  previously  turned  so  as  to  connect  the  manom- 
eter neck  with  the  open  air. 

(2)  Turn  the  stopcock  vS  at  the   base   of   the  manometer 
tube  so  as  to  isolate  the  tube  leading  to  the  neck. 

(3)  Turn  the  stopcock  6*  so  as  to  drain  the  above  tube  into 
a  clean  beaker,  taking  care  to  close  5  at  the  moment  the  level 
of  the  mercury  reaches  the  mark  a.     The  mercury  which  has 
escaped  measures  the  contents  of  the  neck. 

The  computation  of  the  coefficient  of  expansion  of  the  air 
within  the  bulb  follows  readily  from  these  measurements  by  the 
application  of  the  laws  of  Boyle  and  Gay  Lussac.  Thus  : 

A  gas  possessing  a  coefficient  of  expansion  a,  when  heated 
from  o°  to  T°,  expands  from  F0  to  VT,  according  to  the  law 

Fr  =  (i+«7)F0.  (91) 


PROPERTIES   OF   GASES.  ^05 

If  reduced,  when  at  the  higher  temperature,  to  a  volume  V 
by  the  pressure  PT,  we  have 

V  P 

•  rfe-=  v*p;  (92) 

where  P0  is  the  pressure  at  which  the  gas  is  measured  while 
still  at  o°. 

Under  the  conditions  of  the  present  experiment,  we  have 
two  volumes  of  gas  to  consider :  that  within  the  bulb,  which 
has  a  volume  V§  at  o°  and  of  V$(i  +kT)  at  T°,  and  that  within 
the  neck,  the  volume  of  which  is  z>0(i  +kt).  The  full  expression 
for  the  relations  between  volumes,  pressures,  and  temperatures 
therefore  is 

T  _i_  £  T  T  J_  bt\  /  r  -I-  bt\ 

o.  (93) 


in  which  equation  k  is  the  cubical  coefficient  of  the  expansion 
of  the  bulb,  and  z/0  is  the  volume  of  the  neck  and  connecting 
tube  at  the  temperature  o°. 

In  equation  93  we  may  ignore  the  influence  of  temperature 
upon  the  volume  of  the  neck,  and  write  for  VQ(I  +kt)  the  simpler 
form  v  (volume  of  the  neck  at  temperature  of  the  room  t). 
Equation  93  then  becomes 


which  may  be  written 

v«(i+kT}+vl-      P 

•     .  (95) 


PT  and  PQ  are  quantities  obtained  by  adding  the  atmospheric 
pressure  H^  and  H^  observed  respectively  at  the  times  of  the 
first  and  last  adjustment  of  the  manometer  and  the  corresponding 
manometric  pressures  h^  and  h^  (which  may  be  positive  or  neg- 
ative). 


106  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

To  solve  equation  95  we  assume  the  value  a  =  0.003665  for 
the  right-hand  member,  an  approximation  to  our  final  actual 
value  which  will  in  no  appreciable  manner  influence  the  result, 
since  it  enters  only  into  the  correction  of  the  very  small  quan- 
tity v.  The  accuracy  of  the  determination  of  a  depends  upon 
our  knowledge  of  the  quantities  PT,  PQ,  T,  V^  and  k* 


*  For   a   description   of  Regnault's   classical   research   upon   this  subject,  see 
Memoires  de  1'Academie  Royale  des  Sciences,  XXI  (1847). 


CHAPTER   III. 
GROUP    I:    CALORIMETRY. 

(I)  General  statements ;  (1^  Heat,  of  vaporization;  (I2)  Heat  of 
fusion;  (I3)  Specific  heat ;  (I4)  Radiating  and  absorbing 
power. 

(I).     General  statements  concerning  calorimetry. 

It  may  be  said  in  general  that  calorimetric  determinations 
are  subject  to  a  great  variety  of  annoying  errors,  which  can  be 
avoided  only  by  the  exercise  of  especial  care  and  patience 
on  the  part  of  the  experimenter.  The  student  is  therefore 
advised  to  plan  his  work  very  carefully  before  beginning  the 
experiment  itself,  so  that  he  shall  run  no  risk  of  omitting 
essential  observations  and  precautions.  It  will  generally  be 
found  that  the  greatest  source  of  error  in  calorimeter  experi- 
ments is  the  inaccurate  determination  of  temperatures.  This 
may  be  due  to  several  causes  : 

(1)  The  thermometer  may  indicate  the  temperature  of  a 
portion  of  the  liquid ;  the  rest  of  the  liquid  being  at  a  different 
temperature. 

(2)  The  thermometer  may  not  have  had  time  to  acquire 
the  temperature  of  the  surrounding  liquid. 

(3)  The  thermometer  itself  may  be  inaccurate. 

(4)  The  reading  of  the  thermometer  may  be  at  fault. 
These   sources   of   error   should   be  guarded    against   with 

especial  care. 

The  equations  required  for  the  computation  of  results  in 
calorimetry  may  all  be  derived  from  one  general  principle. 

107 


108  JUNIOR   COURSE    IN   GENERAL   PHYSICS. 

This  principle  may  be  stated  as  follows  :  The  amount  of  heat 
lost  by  one  system  of  bodies  is  equal  to  the  amount  gained 
by  another  system.  This,  of  course,  treats  potential  energy 
due  to  change  of  state  as  latent  heat.  The  heat  lost  or  gained 
by  a  body  may  be  due  to  two  causes : 

(1)  Change  in  temperature ;    the  amount  in  this    case  is 
equal  to  the  continued  product  of  the  mass,  specific  heat,  and 
change  in  temperature  of  the  body. 

(2)  Change  of  state ;  this  amount  is  equal  to  the  product 
of    the    mass    so    changed    by   a    constant    quantity    of    heat 
necessary  to  produce  such  a  change  in  unit  mass. 

The  amount  of  heat  lost  by  radiation  to  the  air  cannot 
be  expressed  in  either  of  these  ways ;  but  it  may  be  expressed 
as  equal  to  the  product  of  the  time  during  which  radiation 
takes  place,  the  average  difference  of  temperature  between 
the  radiating  body  and  the  air,  and  the  radiation  constant 
of  the  body. 

I. 

Comparison  of  thermometers. 

When  two  or  more  thermometers  are  used  in  an  experiment, 
their  indications  should  always  be  compared,  to  determine 
whether  their  indications  agree.  Even  the  best  thermometers 
are  apt  to  differ  in  "zero  point,"  so  that  they  may  give  different 
readings  for  the  same  temperature,  and  yet  measure  differences 
in  temperature  accurately. 

To  compare  thermometers,  they  should  be  placed  together  in 
a  vessel  of  water  (at  any  convenient  temperature),  and  alternate 
readings  taken  for  several  minutes,  the  water  being  kept 
thoroughly  stirred.  If  they  are  found  to  differ,  a  suitable 
correction  must  be  made  to  all  subsequent  readings. 

The  numbers,  or  other  distinguishing  marks,  of  the  ther- 
mometers used  should  in  all  cases  be  recorded. 


CALORIMETRY. 


II. 

Determination  of  the  water  equivalent  of  a  calorimeter. 

When  a  calorimeter  containing  water,  etc.,  is  heated  or 
cooled,  heat  is  absorbed  or  given  out  by  the  vessel  itself  in 
addition  to  that  absorbed  or  liberated  by  its  contents.  The 
water  equivalent  of  a  calorimeter  is  a  quantity  of  water  which 
would  absorb  the  same  amount  of  heat,  when  warmed  through 
a  certain  number  of  degrees,  as  is  absorbed  by  the  calorimeter 
when  heated  through  the  same  range  of  temperature. 

To  determine  the  water  equivalent,  proceed  as  follows  : 

(i)  Fill  the  calorimeter  nearly  three-fourths  full  of  water 
three  or  four  degrees  colder  than  the  air,  the  weight  of  the  water 
being  known.  This  water  should  be  kept  thoroughly  stirred, 
and  its  temperature  should  be  observed  by  means  of  a  thermom- 
eter hanging  in  it. 

Add  enough  hot  water,  of  known  temperature,  to  fill  the 
calorimeter  to  within  one  or  two  centimeters  of  the  top. 
Stir  thoroughly,  and  record  the  reading  of  the  thermometer 
in  the  mixture  at  intervals  of  half  a  minute,  until  the 
temperature  becomes  practically  constant.  The  hot  water 
should  be  stirred  immediately  before  it  is  poured  in,  and  the 
temperature  of  both  hot  and  cold  water  should  be  observed 
just  the  instant  before  mixing.  It  is  best  to  choose  the  tem- 
perature of  the  hot  water  so  that  the  mixture  will  come  to  about 
the  temperature  of  the  air,  corrections  for  radiation  being  un- 
necessary if  this  is  done.  The  mass  of  the  hot  water  used 
may  be  determined  by  weighing  the  mixture  after  the  obser- 
vations are  completed.  From  the  data  obtained,  the  water 
equivalent  is  to  be  computed.*  The  student  should  make 
at  least  three  determinations. 

*  The  amount  of  heat  that  the  calorimeter  absorbs  is  very  small  compared  with 
the  amount  absorbed  by  the  water  which  it  contains.  For  this  reason  slight  errors  of 
observation  will  generally  cause  a  very  great  error  in  the  computed  result.  A  common 
source  of  error  is  the  following :  while  the  hot  water  is  being  poured  into  the  cold 


110  JUNIOR   COURSE    IN   GENERAL   PHYSICS. 

If  the  material  from  which  the  calorimeter  is  made  is  known, 
the  water  equivalent  may  also  be  computed,  as  a  check  on  the 
above  results,  from  the  mass  and  specific  heat. 

In  the  determination  of  the  water  equivalent,  great  care 
must  be  used  in  all  temperature  readings,  or  the  results  of 
successive  determinations  will  be  discordant.  This  is  especially 
true  in  the  case  of  small  calorimeters.  To  obtain  the  best  re- 
sults, a  number  of  separate  determinations  should  be  made,  and 
the  average  of  all  the  results  used.  No  single  result  should  be 
discarded  merely  because  it  differs  widely  from  the  rest.  A 
result  can  be  legitimately  discarded  only  when  something  has 
occurred  during  the  experiment  which  tends  to  throw  discredit 
on  some  of  the  observations,  or  when  there  is  an  obvious  mistake 
in  one  of  the  readings. 

In  the  most  accurate  calorimetric  experiments  it  is  necessary 
to  determine  not  only  the  water  equivalent  of  the  calorimeter, 
but  also  the  water  equivalents  of  the  thermometers,  stirring- 
rods,  etc.  In  the  experiments  which  follow,  however,  this  is 
unnecessary. 

In  all  calorimetric  experiments,  the  temperatiire  of  the  room 
should  be  recorded,  as  it  will  be  found  necessary  in  making 
corrections  for  radiation. 

III. 

Determination  of  the  radiation  constant  of  a  calorimeter. 

The  loss  of  heat  from  a  body  which  is  a  few  degrees  warmer 
than  its  surroundings  is  proportional :  (i)  to  the  time  during 
which  radiation  takes  place ;  (2)  to  the  difference  in  temperature 
between  the  body  and  the  room  ;  (3)  to  a  constant  called  the 
constant  of  radiation,  depending  upon  the  nature  and  extent 
of  the  radiating  surface. 


water,  it  will  lose  some  heat  to  the  air.  In  the  computations  this  small  quantity  of 
heat  is  necessarily  treated  as  if  it  were  absorbed  by  the  calorimeter,  thus  giving  too 
large  a  value  to  the  water  equivalent. 


CALORIMETRY.  1 1 1 

Note  that  this  constant  depends  only  on  the  surface,  and 
not  upon  the  nature  of  the  interior  of  the  body.  The  radiation 
constant  of  a  calorimeter  is,  for  example,  the  same  when  it 
contains  mercury  as  when  it  is  filled  with  water.  But  the  rate 
of  cooling  will  be  different  in  the  two  cases  on  account  of  the 
difference  in  the  two  specific  heats.  Radiation  is  essentially 
a  phenomenon  which  occurs  at  the  surface  of  a  body,  and 
depends  wholly  upon  the  nature  and  temperature  of  this 
surface. 

The  gain  of  heat  by  absorption  when  the  body  is  colder 
than  its  surroundings  obeys  the  same  laws.  The  law  above 
stated  is  known  as  Newton's  law  of  cooling,  and  is  really 
only  an  approximation  to  the  truth.  In  the  case  of  bodies 
differing  in  temperature  from  their  surroundings  by  not  more 
than  10°,  the  approximation  is,  however,  good. 

The  radiation  constant  may  be  defined  as  the  amount  of  heat 
which  is  lost  by  radiation  in  one  minute  when  the  radiating 
body  is  one  degree  hotter  than  the  air.  For  a  difference  in 
temperature  of  0°,  the  radiation  is  6  times  as  great;  and  for 
/  minutes  instead  of  one  minute  the  loss  is  t  times  as  great. 
It  will  thus  be  seen  that  if  the  radiation  constant  is  known, 
the  loss  of  heat  from  a  body  such  as  a  calorimeter  can  be 
readily  computed. 

In  most  calorimetric  work,  corrections  must  be  made  for 
the  loss  of  heat  by  radiation,  or  the  gain  by  absorption,  during 
the  time  of  the  experiment.  The  first  step  in  any  calorimetric 
experiment  should  therefore  be  the  determination  of  the  radia- 
tion constant.  The  method  is  as  follows  : 

(i)  Fill  the  calorimeter  to  within  I  or  2  cm.  of  the  top  with 
water  considerably  warmer  than  the  air  (say  io°-2O°  warmer). 
The  mass  of  the  water  should  be  known.  Suspend  a  thermom- 
eter in  the  center  of  the  calorimeter,  and  observe  the  tempera- 
ture at  intervals  of  one  minute  as  the  water  cools.  These 
observations  should  be  continued  for  at  least  an  hour,  the  water 
being  thoroughly  stirred  before  each  reading.  The  temperature 


112  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

of  the  room,  as  indicated  by  a  thermometer  hanging  near, 
should  also  be  occasionally  recorded. 

(2)  With  the  data  obtained  plat  two  curves,  using  times  as 
abscissas  in  each  case,  and  temperatures  of  air  and  water  as 
ordinates.  A  smooth  curve  should  now  be  drawn  in  each  case, 
passing  as  nearly  as  possible  through  all  the  points  platted. 
Any  slight  deviations  from  such  smooth  curves  are  probably 
due  to  accidental  errors  in  the  observations. 

From  the  data  given  by  these  new  curves,  and  knowing  the 
mass  of  water,  the  statements  made  above  may  be  verified,  and 
the  radiation  constant  computed. 

It  should  be  observed  that  an  approximation  must  here  be 
made,  viz.,  that  the  temperature  of  the  surface  of  the  calorim- 
eter is  the  same  as  that  of  the  liquid  contained  in  it.  If  the 
liquid  is  kept  thoroughly  stirred,  and  if  the  material  from  which 
the  calorimeter  is  made  is  a  good  conductor,  no  great  error  is, 
however,  introduced. 

For  example,  let  the  mass  of  water  plus  the  water  equivalent  of  the 
calorimeter  be  500  grams.  Suppose  that  the  temperature  fell  from 
30°  to  28°  in  five  minutes,  the  temperature  of  the  room  being  20°. 
The  temperature  of  the  water  having  changed  2°,  the  loss  of  heat  is 
equal  to  2  x  500,  or  1000  calories.  Since  this  loss  took  place  in  five 
minutes,  the  loss  in  one  minute  was  iooo-f-  5,  or  200  calories.  The 
average  difference  in  temperature  between  water  and  air  was  9°.  The 
loss  for  one  minute,  and  for  i°  difference  in  temperature,  would  there- 
fore be  200  -7-9  =  22  +  minor  calories,  which  is  the  radiation  constant. 
Similar  computations  made  with  different  portions  of  the  data  should 
give  nearly  the  same  result.  Make  eight  or  ten  such  computations  and 
use  the  mean. 

In  using  the  constant  thus  obtained  to  correct  for  radiation  losses, 
it  usually  happens  that  the  temperature  of  the  calorimeter  does  not 
remain  constant  throughout  the  experiment,  so  that  the  rate  at  which 
heat  is  lost  by  radiation  is  continually  changing.  The  method  to  be 
used  in  such  cases  is  illustrated  by  the  following  example  : 

Suppose  that  the  temperature  of  the  calorimeter  is  observed  at 
intervals  of  one  minute  and  is  found  to  vary  as  follows  :  29°,  26. °5, 
24°,  22.°6,  2i.°4,  20.°8,  20.°6,  20.°5,  the  temperature  of  the  air  being 


CALORIMETRY. 


22°.  The  average  temperature  of  the  calorimeter  during  the  seven 
minutes  is  therefore  23.° 1 8,  (found  by  adding  all  the  readings  and 
dividing  by  8) .  Radiation  has  therefore  taken  place  for  seven  minutes 
at  a  rate  whose  average  value  is  that  corresponding  to  a  difference 
in  temperature  of  i.°i8  from  the  air.  If  the  radiation  constant  is  20, 
the  loss  of  heat  is  20  x  1.18  x  7  =  165.2  calories. 

EXPERIMENT  lv     Determination  of  the  heat  of  vaporization 
of  water. 

The   apparatus   for   this   experiment   may  be   arranged   in 
a  great  variety  of  ways.     The  essential  parts  are : 

(1)  Some  vessel  in  which  steam  may  be  generated. 

(2)  A  calorimeter,   which   may  be   any  metallic  vessel  of 
suitable  size. 

(3)  Tubes  of  metal  or  glass  by  which  the  steam  may  be 
conveyed  to  the  calorimeter.     The  latter  should  be  sheltered 


11  il 

KMU 


Fig.  43. 

from  the  heat  radiated  from  the  boiler,  and  some  device  should 
be  supplied  to  prevent  the  water  which  condenses  in  the  tubes 
from  entering  the  calorimeter.  Fig.  43  shows  a  convenient  form 
of  apparatus  for  this  determination. 

The  water  equivalent  and  radiation  constant  of  the  calorim- 


114  JUNIOR   COURSE   IN    GENERAL   PHYSICS. 

eter  used  should  first  be  determined  as  previously  described. 
Observations  may  then  be  made  as  follows  to  determine  the 
heat  of  vaporization. 

(1)  Fill  the  calorimeter  to  within  2  or  3    cm.    of   the   top 
with  a  known  mass  of  water  considerably  colder  than  the  air 
(from  8°  to  12°  colder). 

(2)  Pass  steam  into  the  calorimeter  from  a  vessel  of  boiling 
water  by  means  of  the  tubes  provided  for  the  purpose,  keeping 
the  water  in  the  calorimeter   thoroughly  stirred,  and  observe 
its  rise  in  temperature  at  intervals  of  one  minute,  until  it  has 
been  heated  as  far  above  the  temperature  of  the  room  as  it  was 
previously  below  it. 

(3)  Determine  the  mass  of  steam  condensed  by  weighing 
the  calorimeter  and  contents  at  the  end  of  the  experiment,  the 
weight  of  the  vessel  and  of  the  cold  water  having  been  previ- 
ously determined.     These  weighings  should  be  made  with  con- 
siderable care,  as  the  mass  of  the  condensed  steam  may  be  quite 
small.    To  make  sure  that  the  steam  is  dry,  it  should  be  slightly 
superheated  by  a  flame  placed  under  the  tube  which  leads  to 
the   calorimeter.     The  temperature  of  the  steam   just  before 
entering  the  water  may  be  observed  by  means  of  a  thermometer 
inserted  in  the  tube.     The  steam  should  be  allowed  to  pass 
through  the  tubes  for  a  considerable  time  before  beginning  the 
experiment,   in  order  to  make  sure  that  they  are  thoroughly 
warmed  (to  avoid  condensation). 

(4)  From  the  data  obtained  compute  the  heat  of  vaporization 
of  water,  or  the  latent  heat  of  steam.     Corrections  should  be 
made  for  the  loss  or  gain  of  heat  due  to  radiation  and  absorp- 
tion, and  for  the  heat  capacity  of  the  calorimeter  itself. 

This  correction,  due  to  radiation,  may  be  reduced  to  a 
minimum  by  allowing  the  flow  of  steam  to  continue  until  the 
water  in  the  calorimeter  reaches  a  temperature  as  much  above 
that  of  the  air  as  it  was  initially  below  that  temperature.  But 
the  correction  should  always  be  computed.  At  least  three 
determinations  should  be  made. 


CALORIMETRY. 


The  following  tables  show  the  character  of  the  data  per- 
taining to  this  experiment  and  the  method  of  arranging  them. 


COMPARISON  OF  THERMOMETERS. 


No.  12975. 

No.  12319. 

No.  3. 

7-53 

7-55 

8-3 

7.60 

7.62 

8-3 

26.80 

26.82 

27.4 

25-83 

25-87 

27.2 

38.03 

38.08 

39. 

36-89 

36.91 

37.6 

No.  12327  registers  99°  in  boiling  water. 


RADIATION  CONSTANT. 


Time. 

Tern,  of 
Vessel. 

Tern,  of 
Room. 

Radiation 
Constant. 

Time. 

Tern,  of 
Vessel. 

Tern,  of 
Room. 

Radiation 
Constant. 

3-34 

30.8 

II.  2 

3-47 

28.0 

II.  I 

35 

30.56 

48 

27.8 

36 

30.36 

49 

27.6 

4.15 

37 

30.12 

50 

27.4 

38 

29.90 

"•3 

5i 

27.23 

39 

29.70 

3.98 

52 

27.03 

40 

29.46 

53 

26.88 

II.  0 

4i 

29.28 

54 

26.70 

3-93 

42 

29.03 

55 

26.50 

43 

29-83 

II.  2 

56 

26.33 

44 

28.60 

4.22 

57 

26.15 

45 

28.40 

58 

25-95 

10.8 

46 

28.20 

59 

25.80 

4.l6 

Mass  of  Calorimeter  +  Water  =  492.5  grams. 
Mass  of  Calorimeter  =154.7 


Water  Equivalent 


Radiation  Constant 


337-8 
=    14.8 


352.6  grams. 
=      4.09  calories. 


i6 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


WATER  EQUIVALENT  OF  CALORIMETER. 


.. 

II. 

III. 

Mass  of  Calorimeter, 

154.7 

!54-7 

154.7 

Cal.  +  Cold  Water, 

334-0 

344- 

340.2 

"       Cold  Water, 

179-3 

189.3 

185.5 

"       Cal.  +  Mixture, 

478.5 

480.5 

486.5 

Warm  Water, 

H4.5 

136-5 

146.3 

Tern,  of  Room,  No.  3, 

24.0 

21.  0 

21.0 

"       Cold  Water,  No.  12975, 

9.8 

10.2 

8.25 

"       Warm  Water,  No.  12319, 

35-6 

36.6 

37-4 

"       Mixture,  No.  12975, 

20.88 

20.85 

20.4 

Water  Equivalent, 

12.6 

12.6 

19.2 

Water  equivalent  =  14.8. 

HEAT  OF  VAPORIZATION  OF  WATER. 


I. 

II. 

III. 

Mass  of  Calorimeter, 

154-7 

!54-7 

J54-7 

«       Cal.  +  Cold  Water, 

438.0 

427.7 

434-o 

"       Cold  Water, 

283-3 

273.0 

279.3 

"       Cal.  +  Mixture, 

450.0 

440.6 

447.2 

"       Condensed  Steam, 

12.0 

12.9 

13.2 

Temperature  of  Room, 

21.0 

21.0 

21.0 

Cold  Water, 

10.4 

8.27 

10.82 

No.  12319, 
{  TIME. 

IOS0 

105° 

103° 

"        «         Steam,                   i£m. 

107 

105 

104° 

No.  12327,           3 

108 

105° 

[    4 

107 

o 

13-4 

14.0 

12.8 

5 

16.2 

19.6 

13.6 

I.O 

1  8.0 

27.2 

.  17.8 

"         "         Mixture,              1.5 

20.7 

344 

22.0 

2.O 

23-7 

35-0 

24.7 

No.  12319,         2.5 

26.5 

35-45 

28.5 

3-0 

31-0 

32.9 

3-5 

34-4 

37-0 

4.0 

34-9 

37-7 

Heat  of  Vaporization, 

545 

544 

540 

CALORIMETRY.  H7 

EXPERIMENT  I2.     Determination  of  the  heat  of  fusion  of  ice. 

The  radiation  constant  and  the  water  equivalent  of  the 
calorimeter  used  are  first  to  be  determined,  as  previously 
described.  Observations  may  then  be  taken  to  determine 
the  heat  of  fusion  as  follows : 

(1)  Fill  the  calorimeter  to  within  2  or  3  cm.  of  the  top 
with  a  known  mass  of  water,  3°  or  4°  warmer  than  the  air. 

(2)  Stir  thoroughly  and  observe  the  temperature.    Then  drop 
in  a  piece  of  ice ;    hold  it  under  water  by  means  of  a  stirrer 
arranged  for  the  purpose,  and  observe  the  temperature  of  the 
water  at  intervals  of  half  a  minute  until  the  ice  is  melted,  and 
a  fairly  constant  temperature  is  reached.     In  case  the  melting 
of  the  ice  cools  the  calorimeter  below  the  temperature  of  the 
room,  it  is  well  to  continue  observations  of  temperature,  stirring 
thoroughly  before  each   reading,  until  the  calorimeter  begins 
to  warm  again  by  absorption  of  heat  from  the  air. 

The  ice  used  should  be  at  its  melting-point.  This  is  assured 
by  keeping  it  for  some  time  inside  the  warm  room.  It  should 
be  carefully  dried  by  means  of  filter  paper  just  before  dropping 
in  the  calorimeter. 

The  mass  of  ice  used  may  be  obtained  by  weighing  the 
calorimeter  and  contents  after  the  observations  are  completed, 
the  weight  of  the  vessel  and  of  the  warm  water  being  already 
known. 

From  the  data  obtained  compute  the  heat  of  fusion. 

Corrections  are  to  be  made  for  the  loss  of  heat  by  radiation, 
and  for  the  water  equivalent  of  the  calorimeter.  Make  at  least 
three  complete  determinations. 

EXPERIMENT  I3.  Determination  of  the  specific  heat  of 
a  solid. 

(i)  Place  the  metal  whose  specific  heat  is  to  be  determined 
in  the  calorimeter,  and  support  it  in  such  a  way  that  it  does 
not  touch  the  sides  or  bottom.  Enough  water  of  known  weight 


Il8  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

should  now  be  placed  in  the  calorimeter  to  just  cover  the  metal, 
the  temperature  of  the  water  being  from  8°  to  15°  above  that 
of  the  air. 

(2)  Allow  the  calorimeter  and  contents  to  stand  for  at 
least  ten  minutes  in  order  to  make  sure  that  the  metal  has 
acquired  the  temperature  of  the  water.  Then  add  cool  water, 
stir  thoroughly,  and  observe  the  temperature  at  half-minute 
intervals  until  it  reaches  a  practically  constant  value.  The 
temperature  and  amount  of  the  cold  water  should  be  such  as 
to  bring  the  final  temperatures  of  the  mixture  very  close  to 
that  of  the  air.  A  few  preliminary  trials  will  show  about  what 
the  temperature  should  be.  The  temperature  of  hot  and  cold 
water,  each  thoroughly  stirred,  should  be  observed  immediately 
before  mixing.  The  weight  of  the  cold  water  added  is  to  be 
found  by  weighing  the  calorimeter  and  contents  after  the  other 
observations  are  completed. 

As  the  specific  heat  of  any  metal  is  much  less  than  that 
of  water,  it  will  be  advisable  to  take  a  rather  large  mass  of 
the  metal.  For  good  results,  its  heat  capacity  should  be 
comparable  with  that  of  the  mass  of  water  used.  If  the  metal 
is  not  a  good  conductor  of  heat,  it  should  be  in  small  pieces. 

The  method  here  described  is  merely  one  of  many  which 
may  be  used  in  the  determination  of  specific  heat. 

The  student  will  find  it  instructive,  if  time  is  available,  to 
check  his  results  by  one  of  the  numerous  other  methods  which 
will  be  found  described  in  various  text-books. 

The  water  equivalent  and  the  radiation  constant  of  the 
calorimeter  used  are  to  be  determined  as  described  in  the 
general  directions  at  the  beginning  of  this  group. 

The  weight  of  the  metal  being  known,  its  specific  heat 
may  now  be  computed.  Corrections  are  to  be  made  for 
radiation  and  for  the  absorption  of  heat  by  the  calorimeter 
itself.  At  least  three  determinations  should  be  made. 


CALORIMETRY.  II9 

EXPERIMENT  I4.  Radiating  and  absorbing  powers  of  dif- 
ferent surfaces. 

The  objects  of  this  experiment  are  to  investigate  the  radia- 
tion and  absorption  of  heat  from  different  surfaces,  and  to 
determine  the  relation  between  the  radiating  and  absorbing 
powers  of  the  same  surface. 

The  radiating  constant  of  a  surface  may  be  defined  as  the 
number  of  calories  that  will  be  radiated  from  one  square  centi- 
meter of  the  surface  in  one  minute,  for  a  difference  in  tempera- 
ture of  one  degree  between  the  surface  and  its  surroundings. 
In  like  manner  the  constant  for  absorption  may  be  defined  as 
the  number  of  calories  that  will  be  absorbed  by  one  square 
centimeter  of  the  surface  under  similar  conditions. 

The  radiation  constant  of  a  surface  may  be  determined  by 
dividing  the  heat  lost  by  a  vessel  in  a  given  time,  by  the  time, 
the  average  difference  in  temperature  between  the  surface  and 
the  air,  and  the  area  of  the  vessel.  The  absorption  constant 
may  be  computed  in  a  similar  manner  from  the  heat  gained  in 
a  given  time. 

It  is  to  be  observed  that  radiation  and  absorption  depend 
upon  the  temperature  of  the  radiating  or  absorbing  surface,  and 
not  upon  the  temperature  of  the  contents  of  the  vessel.  If 
the  walls  of  the  vessel  are  thin,  however,  and  of  some  highly 
conducting  material,  no  great  error  is  introduced  by  assuming 
that  the  contents  of  the  vessel  are  at  the  same  temperature 
as  the  surface. 

The  method  of  the  experiment  is  as  follows  : 

(1)  Fill  the  vessel  for  whose  surface  the  radiation  constant 
is  to  be  determined  with  water  1 5°  or  20°  warmer  than  the  air, 
and  place  it  upon  a  poorly  conducting  support,  such  that  the 
vessel  will  be  free  to  radiate  its  heat  in  all  directions. 

(2)  Observe  the  temperature  by  means  of  a  thermometer  hang- 
ing in  the  center  of  the  vessel,  at  intervals  of  two  minutes,  stirring 
the  water  thoroughly  before  each  reading.      The  temperature 
of  the  air  should  also  be  observed  at  intervals  of  about  five 


120  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

minutes,  and  for  good  results  must  remain  nearly  constant 
throughout  the  experiment.  Continue  these  observations  for 
at  least  half  an  hour. 

A  curve  should  now  be  platted  with  times  as  abscissas  and 
temperatures  as  ordinates.  From  this  curve,  or  from  the  data 
themselves,  make  four  or  five  independent  computations  of  the 
radiation  constant.  If  the  constant  is  computed  from  the  curve, 
it  will  be  necessary  to  find  the  "pitch,"  dt-^dT  (^tempera- 
ture; T  =  time),  at  different  points  on  the  curve,  by  drawing 
tangents. 

From  Newton's  law  of  cooling,  the  radiation  constant  R  is 
given  by  the  equation 


where  c  is  the  heat  capacity  of  the  vessel,  A  its  superficial  area, 
and  4  the  temperature  of  the  air.  The  value  of  c  is  deter- 
mined by  adding  the  water  equivalent  of  the  vessel  to  the 
weight  of  the  water  contained  in  it. 

The  following  method  of  computing  the  results  will  be 
found  instructive  as  an  example  of  the  employment  of  graphical 
methods,  and  may  be  used  instead  of  the  above  if  desired. 

).  (97) 

(98) 


By  integration  :  log  (/-  4)  =         T  +  K,  (99) 

where  K  is  the  constant  of  integration. 

If,  therefore,  a  curve  is  platted  whose  co-ordinates  are  T  and 
log(/—  fa)  respectively,  the  result  should  be  a  straight  line.    In  the 

RA 
equation  of  this  line,  -  -  enters  as  one  of  the  constants.     Note 

that  the  logarithm  which  occurs  in  the  above  equation  is  the 


CALORIMETRY.  12I 

Napierian  logarithm.  Ordinary  logarithms  may,  however,  be 
used  until  the  final  result  is  reached. 

The  constant  for  absorption  can  be  determined  in  a  similar 
manner  by  filling  the  vessel  with  water  15°  or  20°  colder  than 
the  air,  and  observing  the  gradual  rise  in  temperature  due  to 
absorption. 

These  observations  should  be  repeated  with  three  or  four 
vessels  which  are  of  the  same  size  and  shape,  but  differ  widely 
in  the  character  of  the  radiating  surface.  Polished  metal  and 
lampblack  surfaces  will  probably  be  found  to  differ  most  widely 
in  their  radiating  powers.  No  difficulty  should  be  experienced 
in  carrying  on  the  observations  with  four  vessels  at  the  same 
time. 

It  is  to  be  observed  that  a  slight  error  is  introduced  in  this 
experiment  by  assuming  that  all  the  heat  is  lost  by  radiation,  for 
part  of  the  loss  is  really  due  to  convection.  For  small  differences 
of  temperature,  however,  the  loss  by  convection  is  small,  and 
may  be  treated  as  though  it  obeyed  Newton's  law.  The  radia- 
tion constants  obtained  represent,  therefore,  the  sum  of  the 
losses  due  to  the  two  causes. 

(In  connection  with  this  experiment,  see  the  general  direc- 
tions for  calorimetric  work.) 


% 


CHAPTER   IV. 
GROUP  P :   STATIC  ELECTRICITY. 

(P)  General  statements ;  (Pt)  Electrostatic  induction;  (P2)  The 
principle  of  the  condenser;  (P3)  The  Holtz  machine; 
(P4)  Further  experiments  with  the  Holtz  machine. 

(P).   General  statements  concerning  static  electricity. 

Whenever  a  body  or  system  of  bodies  becomes  electrified, 
equal  quantities  of  positive  and  negative  electricity  are  pro- 
duced. 

Many  experimental  facts  lead  to  the  conclusion  that  the 
energy  of  electrification  exists  in  the  insulating  medium  between 
the  bodies  containing  these  two  equal  quantities  of  positive  and 
negative  electricity.  These  experimental  facts  prove  that  the 
insulating  medium  is  in  a  state  of  strain.  Therefore  the  energy 
of  electrification  is  the  potential  energy  of  an  electrical  field,  in 
an  insulating  medium,  bounded  by  bodies  containing  what  are 
called  "charges  of  electricity." 

If  an  electrified  body  or  system  of  bodies  be  placed  within  a 
closed  conducting  surface,  the  charge  of  electricity  on  this  sur- 
face is  equal,  and  of  opposite  sign,  to  the  charge  of  the  body  or 
system  of  bodies.  This  law  has  been  deduced  directly  from 
experiment.  However,  it  may  be  shown  to  be  directly  deducible 
from  the  following  theorem. 

Let  F  denote  the  resultant  electrical  force  at  a  point  on  a 
small  element  of  the  surface  of  a  charged  body :  the  integral  of 
the  quantity  FdA,  taken  over  the  entire  surface  of  the  charged 
body,  is  numerically  equal  to  4  TrQ,  in  which  Q  is  the  number  of 

122 


STATIC   ELECTRICITY. 


123 


units  of  electricity  in  the  body.*     This  is  known  as  Green's 
theorem. 

Another  way  of  stating  this  fact  is  as  follows  :  The  number 
of  lines  of  force,  or  of  unit  tubes  of  force,  issuing  from  the 
surface  of  a  body  charged  with  Q  units  of  electricity,  is  47rQ. 
These  lines  of  force,  or  tubes  of  induction,  must  end  on  some 
other  body  or  bodies.  On  the  surfaces  of  the  conductors  where 
these  47r<2  lines  of  force  end,  there  must  be  Q  units  of 
induced  electricity  of  the  opposite  sign  to  the  electricity  on 
the  first  conductor.! 

To  completely  discharge  a  conductor,  and  cause  to  vanish 
the  field  surrounding  it,  it  will  be  necessary  for  these  two  equal 
quantities  of  electricity  of  opposite  signs  to 
unite. 

The  conception  of  free  and  bound  elec- 
tricity helps  to  the  understanding  of  this 
and  other  phenomena  of  static  electricity. 
The  term  "free  electricity,"  or  "free 
charge,"  is  applied  to  that  portion  of  a 
charge  which  will  escape  to  the  earth,  when 
the  conductor  containing  it  is  connected  to 
earth,  while  a  bound  charge  is  that  portion 
which  is  held  by  the  induction  of  some  other  near-by  insulated 
charge. 

Suppose  A  (Fig.  44)  to  be  an  insulated  conductor  charged 
with  Q  units  of  positive  electricity.  Suppose  B  to  be  a 
conductor  which  has  been  grounded  and  afterwards  insulated. 
The  charge  Q  induces  on  B,  q'  units  and  on  the  walls  of  the 
room,  q"  units  of  negative  electricity,  such  that 

<2= -(?'+*")- 


*  Gray,  Absolute  Measurements  in  Electricity  and  Magnetism,  vol.  I,  p.  10. 

t  This  is  more  general  than  the  second  law  above  given,  but  it  is  based  on  the 
assumption  that  an  electrical  field  does  not  extend  indefinitely  in  a  direction  in 
which  there  are  no  charged  bodies. 


124  JUNIOR   COURSE    IN   GENERAL   PHYSICS. 

That  part  q'  of  the  electricity  induced  on  B  is  bound  by  the 
charge  Q.  None  of  it  will  escape  to  the  earth,  for  its  potential 
has  been  reduced  to  zero  by  grounding  it.  The  charge  q'  on  B 
binds,  by  induction,  a  portion  of  the  charge  on  A  ;  so  that  if 
A  were  grounded,  only  a  portion  of  the  Q  units  of  electricity 
would  escape.  That  which  escapes  is  free  electricity ;  the 
remainder  is  bound  by  the  negative  charge  on  B. 

It  is  very  important  to  keep  clearly  in  mind  the  distinction 
between  the  character  and  the  potential  of  a  charge  of  electricity. 
In  the  above  example,  before  A  was  grounded,  B  was  at  zero 
potential,  but  it  had  a  negative  charge ;  after  A  was  grounded, 
the  potential  of  B  became  negative,  although  its  charge  was 
unchanged.  A,  however,  was  reduced  to  zero  potential,  but  it 
still  retained  a  positive  charge. 

Positive  and  negative  electricity  always  exist  at  the  positive 
and  negative  ends  respectively  of  electrical  lines  of  force ;  or, 
as  some  may  prefer  to  put  it,  at  the  positive  and  negative 
boundaries  of  an  electrical  field  of  force.  The  potential  of  the 
body  containing  the  positive  charge  must  always  be  positive 
with  respect  to  the  body  containing  a  negative  charge  at  the 
other  boundary  of  the  field ;  but  the  potential  of  either  or 
both  of  these  bodies  may  be  anything  with  respect  to  the 
earth,  whose  potential  is  usually  taken  as  zero. 

The  potential  of  a  conductor  is  positive,  when,  upon  being 
grounded,  positive  electricity  is  discharged  to  the  earth ;  when 
negative  electricity  is  thus  discharged,  the  potential  is  negative, 
and  when  no  discharge  occurs,  the  conductor  is  at  zero  potential. 

It  is  a  very  instructive  exercise  to  map  out  a  field  of  force 
with  equipotential  surfaces  and  lines  of  force.*  It  is  not 
difficult  to  do  this  in  an  approximate  manner,  if  the  student 
keeps  clearly  in  mind  the  definitions,  the  fact  that  lines  of 
force  and  equipotential  surfaces  are  mutually  perpendicular, 


*  In  this  connection  the  beautiful  maps  of  the  electrostatic  field  at  the  end  of 
the  first  volume  of  Maxwell's  Electricity  and  Magnetism  should  be  inspected. 


STATIC  ELECTRICITY. 


125 


and  the  fact  that  the  surface  of  every  conductor  is  an  equi- 
potential  surface. 

Let  it  be  required  to  map  a  section  of  the  field  within  a 
hollow  conductor  at  zero  potential,  containing  two  insulated 
conductors.  One  of  these  conductors  is  positively  charged,  and 
the  other  has  only  an  induced  charge. 

It  will  be  found  easier  to  draw  the  lines  of  force  first. 

(1)  They  must   always   be   drawn   between   conductors  of 
different  potentials. 

(2)  They   must    issue   from   a   conductor   at   right   angles 
to  its  surface. 

(3)  Lines  of  force  must  always  issue  from  a  body  containing  a 
positive  charge,  and  end  on  a  body  containing  a  negative  charge. 

If  lines  of  force  are  drawn  fulfilling  these  conditions, 
they  will  be  as  indicated  in  Fig  45.*  It  may  be  assumed, 
approximately,  that  along  the  shortest  distance  between  the 
two  conductors  the  potential  falls  uniformly.  Assume  that  the 
difference  of  potential  between  them  is  nine.  Divide  the  distance 
into  nine  equal  parts,  and  through  each  point  of  division  draw  a 
line,  and  continue  it  so  that  it  is  everywhere  perpendicular  to 
lines  of  force.  Each  of  these  lines  must  be  a  closed  curve.  And 
from  definition,' each  of  them  must  lie  in  an  equipotential  surface. 

By  definition,  the  same  amount  of  work  is  done  in  carrying 
a  charge  from  one  point  in  an  equipotential  surface  to  any 

*  The  equipotential  lines  and  lines  of  force  in  Fig.  45  were  computed  by  C.  D. 
Child. 

This  computation  was  made  as  follows :  Known  charges  were  supposed  to  be 
concentrated  at  points.  A  series  of  points  were  then  found  which  had  the  same 

potential,  according  to  the  formula  V  =  2  -^ .     All  the  equipotential  surfaces,  from  3 

K 

to  15  inclusive,  were  determined  in  this  way.  A  conductor  connected  to  the  ground 
was  then  supposed  to  coincide  with  the  equipotential  surface  3.  This  reduced  the 
potential  of  every  point  within  by  three  units.  A  conductor  was  then  supposed  to 
surround  a  charge  of  +  40  units,  and  coincide  with  the  equipotential  surface  12; 
while  another  conductor  was  supposed  to  surround  charges  of  +  io>  —  5>  and  —  5 
units,  and  coincide  with  the  equipotential  surface  3.  These  two  conductors  in  no 
wise  changed  the  potential  of  any  point  in  the  field  of  force,  while  it  was  perfectly 
allowable  to  suppose  the  charges  within  them  to  be  transferred  to  their  outside  surfaces. 


126  JUNIOR   COURSE   IN    GENERAL   PHYSICS. 

point  in  another  equipotential  surface.  Therefore,  the  field 
must  be  strongest  where  these  surfaces  are  closest  together. 
Strength  of  field  is  sometimes  represented  by  the  number 
of  lines  of  force  per  square  centimeter.  Therefore,  where  the 


A  is  a  conductor  [Potential  12  ;  charge  40?]. 
B  is  a  conductor  [Potential  3  ;  induced  charge  —  10  q  and  +  I0q~\ 
Fig.  45. 

equipotential  surfaces  are  closest  together,  the  lines  of   force 
should  be  most  numerous. 

It  should  be  noticed  that  one  of  the  conductors  has  lines  of 
force  issuing  from  it  and  also  ending  on  it.  It  follows  that  there 
is  a  positive  and  a  negative  charge  on  this  conductor,  although 
the  whole  of  it  is  at  the  same  potential. 


STATIC   ELECTRICITY.  I2/ 

Nearly  all  experiments  on  static  electricity  are  more  suc- 
cessful in  cold  weather  than  in  warm.  This  difference  is  prob- 
ably due  to  a  difference  in  humidity.  In  moist  air,  bodies  are 
rapidly  discharged,  so  that  it  is  relatively  much  more  difficult 
to  accumulate  charges  when  the  surrounding  atmosphere  is 
moist.  In  cold  weather  the  absolute  humidity  is  usually  much 
less  than  in  warm  weather.  In  an  artificially  heated  room, 
the  absolute  humidity  remaining  the  same  as  the  outside  air, 
the  relative  humidity  is  very  much  lessened  on  account  of  the 
higher  temperature. 

From  the  definition  of  a  line  of  force,  a  positively  charged 
body  (an  insulated  pith-ball,,  for  example)  tends  to  move  along 
the  lines  of  force  from  positively  charged  bodies  towards  nega- 
tively charged  bodies.  If  the  body  were  negatively  electrified, 
it  would  tend  to  move  in  the  opposite  direction.  This  offers 
a  means  of  testing  the  direction  of  lines  of  force,  and  con- 
sequently the  character  of  the  charge  on  a  charged  body. 

If  an  insulated  pith-ball  be  positively  electrified  (by  being 
brought  in  contact  with  a  glass  rod  which  has  been  previously 
rubbed  with  silk),  and  then  be  suspended  in  a  region  supposed 
to  be  a  field  of  electrical  force,  surrounding  a  charged  con- 
ductor, one  of  three  things  will  occur : 

(1)  The  pith-ball  will  tend  to  move  from  the  body  supposed 
to  be  charged.     This  proves  that  the  region  is  an  electrical 
field  with  lines  of  force  issuing  from  the  body,  which  is  there- 
fore positively  charged. 

(2)  The  pith-ball  will  not  tend  to  move  at  all.     In  this  case 
we  infer  that  the  electrical  field  is  too  weak,  or  that  the  charge 
on  the  pith-ball  is  too  weak  to  produce  a  perceptible  effect. 

(3)  The  pith-ball  will   tend  to  move   towards  the  charged 
body.    This  indicates  that  the  region  is  a  field  of  force  with  lines 
of  force  entering  the  body,  which  is  therefore  negatively  electri- 
fied.    We  say  "  indicates,"  for  it  only  proves  that  there  is  now 
a  field  between  the  pith-ball  and  the  body,  and  that  one  of  them 
was  originally  electrified  before  they  were  brought  near  together. 


128  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

We  know  that  if  the  pith-ball  were  originally  neutral,  it  would 
move  toward  a  strongly  charged  body  when  brought  near  it. 
If  the  conductor  were  strongly  charged,  and  the  pith-ball  weakly 
charged,  both  with  positive  electricity,  the  motion  would  be 
the  same.  Moreover,  if  the  conductor  were  neutral,  a  charged 
pith-ball  brought  near  it  would  tend  to  move  towards  it. 

These  facts  may  be  readily  explained  on  the  theory  of  induc- 
tion. What  is  to  be  learned  from  this  is  that  an  electrical  field 
of  force,  and  the  character  of  the  charge  on  a  body,  cannot  be 
certainly  determined  from  the  motion  of  a  charged  pith-ball 
towards  the  body.  Under  these  circumstances,  the  pith-ball 
should  be  charged  negatively  (by  being  brought  in  contact  with 
vulcanite  previously  rubbed  with  fur),  and  again  experimented 
upon. 

Sometimes  it  is  better  to  first  bring  the  pith-ball  into  contact 
with  the  body  supposed  to  be  charged,  and  then  to  test  the 
nature  of  the  charge  on  the  pith-ball  by  bringing  it  near  electri- 
fied glass  and  vulcanite  rods  in  turn. 

The  electrification  of  a  body  may  often  be  tested  more  reli- 
ably by  the  use  of  a  proof -plane  and  a  gold-leaf  electroscope. 
In  using  the  electroscope,  it  must  be  remembered  that  it  is  the 
first  motion  of  the  leaves,  as  the  charged  proof-plane  is  brought 
near  it,  that  is  to  be  noted.  If  the  proof-plane  has  a  considera- 
ble charge,  whose  sign  is  opposite  to  that  of  the  leaves,  it  will 
cause  them  to  collapse,  and  afterwards  to  diverge,  as  it  is 
brought  quite  near  to  the  electroscope. 

EXPERIMENT  Px.     Electrostatic  induction. 

Every  insulated  conductor  in  the  neighborhood  of  a  positively 
charged  body  has  induced  upon  its  surface  equal  quantities  of 
positive  and  negative  electricity.  The  positive  electricity  is  on 
the  side  farthest  from  the  charged  body,  and  the  negative  on 
the  side  nearest. 

The  object  of  this  experiment  is  to  investigate  this  and  other 
phenomena  of  electrostatic  induction. 


STATIC   ELECTRICITY. 


I29 


For  this  experiment,  a  rather  large  insulated  conductor  is 
required.  A  Leyclen  jar,  in  which  the  small  knob  is  replaced 
by  a  sphere  10  or  12  cm.  in  diameter,  and  connected  with  the 
inner  coating,  is  excellent  for  this  purpose  on  account  of  its 
great  capacity.  Such  a  conductor  will  usually  retain  its  charge 
for  the  whole  time  of  the  experiment.  A  second  insulated 
conductor,  preferably  an  elongated  cylinder  with  hemispherical 
ends,  is  also  required. 

Charge  the  sphere  connected  to  the  Leyden  jar  by  means  of 
an  electrical  machine.  Place  the  second  conductor  in  the  elec- 
trical field  produced  by  the  charged  conductor.  The  two  con- 
ductors should  be  not  more  than  2  or  3  cm.  apart. 

The  nature  of  the  charge  on  different  parts  of  the  second 
conductor  should  now  be  investigated.  This  may  be  done, 
either  by  means  of  a  pith-ball  suspended  by  a  silk  fiber,  or  by 
means  of  a  proof-plane  and  gold-leaf  electroscope.  Some  idea 
may  be  formed  of  the  direction  of  the  lines  of  force  in  the 
electrical  field  surrounding  the  conductors  by  the  direction  in 
which  a  positively  charged  pith-ball  tends  to  move. 

After  testing  as  above,  remove  the  conductor,  still  insulated, 
to  a  distance  from  the  charged  body,  and  test  again.  Place  the 
conductor  again  in  proximity  to  the  charged  body,  ground  the 
conductor,  and  test  with  the  pith-ball  as  before.  Then  move 
the  conductor  closer  to  the  charged  body,  and  note  any  change 
in  its  condition.  Finally,  remove  the  conductor  to  a  distance 
from  the  inducing  body  (the  connection  with  the  ground  having 
been  first  broken),  and  test  again. 

Throughout  the  experiment  care  must  be  taken  not  to 
allow  any  discharge  from  the  charged  body  to  the  second 
conductor. 

To  secure  uniform  results  it  will  generally  be  necessary 
to  repeat  these  tests  several  times.  This  experiment  will 
give  satisfactory  results  only  when  the  air  is  rather  dry.  It 
succeeds  best  in  cold  weather,  when  the  room  is  artificially 
heated. 

VOL.  I  —  K 


130  JUNIOR   COURSE    IX    GENERAL   PHYSICS. 

Addenda  to  the  report: 

(1)  Define  the  following  :    unit  electrical  charge  ;   electrical 
field  of  force  ;    field  of  unit  intensity ;    electrical  difference  of 
potential  ;   unit  difference  of  potential ;    electrical  potential  at 
a  point ;  equipotential  surfaces ;    electrical  line  of    force  ;    unit 
line  of  force,  or  unit  tube  of  force  ;    electrical  capacity  ;   unit  of 
capacity. 

(2)  Give  a  demonstration  of  the  fact  that  lines  of  force  and 
equipotential  surfaces  are  mutually  perpendicular. 

(3)  State  the  general  relation  which  the  quantity  of  induced 
electricity  on  the  conductor  bears  to  the  quantity  on  the  charged 
body  and  the  distance  between  them. 

(4)  Assuming  that  the  charge  of  the  charged  body  is  posi- 
tive, what  is  the  potential  of  the  conductor  in  each  of  the  five 
cases  investigated  ?     What  would  be  its  potential  if  the  charged 
body  were  negative  ? 

(5)  Draw  a  vertical  section  of  the  two  conductors  showing 
equipotential  surfaces  and  lines  of  force. 

(6)  Draw  two  such  figures,  one  when  the  conductor  having 
the  induced  charge  is  insulated,  and  one  when  it  is  grounded. 

EXPERIMENT  P^     The  principle  of  the  condenser. 

When  a  conductor  connected  to  the  earth  is  brought  near  a 
charged  body,  the  potential  of  the  charged  body  is  reduced.  If 
the  conductor  almost  surrounds  the  charged  body,  and  is  very 
close  to  it,  its  potential  will  be  very  greatly  reduced,  although 
the  amount  of  the  charge  remains  absolutely  unchanged. 

Another  way  of  viewing  this  fact  is  to  consider  that  the 
conductor  lessens  the  quantity  of  free  electricity  on  the  charged 
body.  The  remainder  of  the  charge  is  bound  by  the  electricity 
induced  on  the  near-by  conductor.  If,  instead  of  maintaining 
the  charge  constant,  the  potential  of  the  charged  body  is  main- 
tained constant,  it  will  be  found  that  the  charge  must  be 
rapidly  increased  as  the  conductor  connected  with  the  earth  is 
brought  very  near. 


STATIC   ELECTRICITY.  !3I 

A  combination  of  two  conductors,  very  close  together,  one 
of  which  is  connected  to  the  earth,  is  called  a  condenser.  The 
capacity  of  such  a  condenser  is  enormously  greater  than  the 
capacity  of  either  of  the  conductors  of  which  it  is  composed 
when  measured  in  the  absence  of  the  other. 

In  order  to  become  familiar  with  the  phenomena  of  the 
condenser,  two  forms  are  to  be  experimented  with  : 

I. 

The  first  form  is  an  apparatus  consisting  of  two  vertical, 
parallel  metal  plates.  These  plates  are  both  insulated,  and  are 
capable  of  motion  along  a  line  joining  their  centers. 

(1)  Fasten  a  pith-ball  by  means  of  a  conducting  thread  to 
one  plate,  so  that  the  ball  rests  against  the  plate. 

(2)  Connect  the  second  plate  to  the  earth,  and  charge  the 
first  one  by  means  of  an  electrical  machine. 

(3)  Move   the   plates   to   and   from   each   other,  and   note 
the  effect  on  the  pith-ball. 

(4)  When  the  plates  are  quite  near  together,  insulate  the 
plate  that  was  formerly  grounded,  and  afterwards  discharge  the 
other  plate  by  grounding  it.     Then  separate  the  plates,  and  note 
the  effect  on  the  pith-ball. 

(5)  Fasten  a  pith-ball,  as  above,  to  the  second  plate  also. 
Charge  the  plates  while  I  or  2  cm.  apart  by  connecting  them 
to  the  opposite  terminals  of  an  electrical  machine.     Insulate 
the  two  plates  without  grounding  either  of  them,  and  determine 
the    character   of    the    charge   on    each   plate,   by   bringing  a 
charged  body  whose    condition  is  known  near   each   pith-ball 
in  turn. 

(6)  Connect  one  of  the  plates  to  the  ground  for  an  instant, 
and  observe  the  effect  on  the  sign  and  magnitude  of  the  charges. 
Do  the  same  with  the  second  plate.     Continue  grounding  alter- 
nately the  two  plates  until  they  are  both  very  nearly  discharged. 

(7)  Charge  the  plates  again,  and  observe  the  effect  of  con- 
necting them  by  means  of  a  good  conductor.     Repeat  these 


132  JUNIOR  COURSE  IN   GENERAL   PHYSICS. 

observations  with  a  glass  plate  between  the  metal  conductors, 
the  latter  being  very  close,  or  in  contact  with  the  glass. 

II 

The  other  form  of  condenser  to  be  experimented  with  is  a 
Ley  den  jar. 

(1)  Place  the  jar  on  an  insulating  support,  and  charge  it  by 
connecting  the  two  coatings  to  the  opposite  terminals  of   an 
electrical  machine. 

(2)  Disconnect  from  the  electrical  machine  without  ground- 
ing either  coating,  and  experiment  as  with  the  plate  condenser. 

(3)  Determine  the  number  of  alternate  groundings  of  the  two 
coatings  necessary  to  reduce  the  charge  to  a  definite  fraction 
of  its  original  value.     For  the  purpose  of  this  determination, 
the  assumption  may  be  made  that  the  charge  is  proportional 
to    the    length    of    spark,  when    either    coating    is    grounded, 
the  other  coating  having  been  previously  grounded  and  then 
insulated. 

(4)  When  the  jar  is  fully  charged,  make  metallic  connection 
between  the  two  coatings.     After  a  few  minutes  connect  the 
coatings  again,  and  note  the  existence  of  the  "residual  charge." 

(5)  Try  to  charge  the  jar  by  connecting  only  one  coating  to 
the  electrical  machine,  the  other  coating  being  insulated.    Investi- 
gate the  nature  of  charges  on  the  two  coatings,  and  afterwards 
discharge  the  jar,  observing  whether  the  spark  is  comparable 
with  that  obtained  when  the  jar  was  charged  by  the  method 
first  given. 

The  above-described  experiments  should  be  repeated  several 
times  in  order  to  be  certain  of  the  results  and  to  become 
familiar  with  the  phenomena. 

Addenda  to  the  report: 

(i)  Indicate  whether  in  the  case  of  a  condenser  with  a 
gas  or  a  liquid  as  dielectric,  there  would  be  anything  com- 
parable to  the  residual  charge  of  a  Leyden  jar. 


STATIC   ELECTRICITY. 


133 


(2)  Indicate  why  it  requires  a  very  large  number  of  alter- 
nate groundings  of   the  two  coatings  of   a  condenser  to  per- 
ceptibly reduce  its  charge. 

(3)  Assume  that  the  alternate  groundings  of  the  two  coat- 
ings are  at  equal  intervals  of  time,  and  draw  two  curves  with 
times  as  abscissas  and  potentials  of  the  two  coatings  as  ordi- 
nates. 

(4)  Draw  a  vertical  section  of  the  jar,  with  coatings  quite 
wide  apart  and  showing  lines  of  force  and  the  vertical  sections 
of  equipotential  surfaces. 

(5)  Draw  two  such  figures,  one  in  which  the  potential  of 
one   coating   is   zero,   and   one   in  which   the  surface  of  zero 
potential  lies  between  the  coatings. 

(6)  Determine  approximately  the  electrostatic  capacity  of 
the  jar  from  its  dimensions. 

(7)  Assume  the  difference  of  potential  between  the  coatings 
to   be    100   electrostatic   units,  and   compute   the   electrostatic 
force  in  the  glass  between  the  coatings. 

(8)  Compute  the  total  charge  in  the  jar  under  the  above 
conditions. 

(9)  Compute  the  energy  of  the  charge. 

EXPERIMENT  P3.     The  Holtz  machine. 

In  all  influence  machines,  mechanical  energy  is  directly 
transformed  into  the  energy  of  electrification.  The  object  of 
this  experiment  is  to  familiarize  the  student  with  the  use  of 
such  machines  and  the  principles  involved  in  their  action.  Any 
type  of  influence  machine  may  be  used.  The  following  is  the 
procedure : 

(i)  Run  the  machine  a  few  seconds  until  it  is  fully  charged. 
The  poles  should  be  a  few  centimeters  apart.  Then  stop  the 
machine,  and  determine,  by  means  of  a  pith-ball,  the  character 
of  the  charge  on  every  part  of  the  machine.  Repeat  these 
observations  several  times,  and  observe  whether  the  polarity  of 
the  machine  becomes  reversed. 


134  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

(2)  While  the  machine  is  charged  and  at  rest,  gradually  bring 
the  terminals  together  until  a  discharge  takes  place,  and  observe 
the  effect  upon  the  pith-ball.     Determine  the  character  of  the 
charges  on  different  parts  of  the  machine  when  it  is  running 
steadily  with  the  terminals  too  far  apart  to  allow  a  discharge. 

(3)  Observe  the  difference  in  the  discharge  when  the  Leyden 
jars  are  removed,  also  when  they  are  replaced  by  larger  ones. 

(4)  Determine  the  maximum  distance  between  the  terminals/ 
at  which  a  discharge  will  pass  when  the  machine  is  running  stead- 
ily, but  not  very  rapidly.     Remove  the  crossbar,  and  determine 
the  maximum  length  of  spark  between  the  terminals  when  the 
machine  is  running  at  the  same  rate  as  before. 

(5)  Reverse  the  direction  of  rotation,  and  determine  under 
what  conditions  the  machine  will  work. 

(6)  Take  the  machine  into  a  dark  room,  and  run  it  steadily 

(a)  with  the  crossbar  in  position,  and  the  terminals  in  contact ; 

(b)  with  the  crossbar  in  position,  and  the  terminals  very  wide 
apart ;    (c)  without  the  crossbar,  and  with  the  terminals  first  in 
contact,  and  afterwards  widely  separated.      Observe   carefully 
the  brush  discharge  between  the  revolving  plate  and  the  combs 
in  all  these  cases. 

Addenda  to  the  report: 

(1)  Indicate   the  results,   by  positive  and   negative  signs, 
upon  carefully  drawn  diagrams  of  the  machine. 

(2)  Explain  how  the  machine  becomes  highly  charged  when 
one  armature  is  given  a  small  initial  charge,  and  the  plate  is 
steadily  revolved. 

(3)  Indicate  the  function   of   the  crossbar,   and   the  most 
advantageous  position  for  it. 

(4)  Indicate  the  function  of  the  Leyden  jars. 

EXPERIMENT  P4.     The  Holtz  machine  (continued}. 
After  performing  Exp.  P3,  the  following  further  experiments 
with  an  electrical  machine  will  be  found  very  instructive  : 


STATIC   ELECTRICITY. 


I. 


135 


Remove  the  Leyden  jars,  and  connect  to  each  terminal 
of  the  machine  one  coating  of  a  condenser  whose  capacity  may 
be  varied  in  a  known  manner.  Connect  together  the  remaining 
coatings  of  the  two  condensers.  Condensers  formed  by  coating 
the  whole  of  one  side  of  a  glass  plate  with  tin-foil,  while  on  the 
other  side  are  several  pieces  of  tin-foil  insulated  from  each 
other,  and  of  equal  area,  serve  very  well  for  this  purpose. 

Place  the  terminals  at  a  fixed  distance  apart  of  2  or  3  cm., 
and  run  the  machine  uniformly,  counting  the  number  of  dis- 
charges per  minute.  Vary  the  capacity  of  the  condensers 
connected  to  the  terminals,  and  repeat  these  observations. 

If  the  machine  works  uniformly,  it  will  be  found  that  the 
number  of  sparks  per  minute  varies  inversely  as  the  capacity 
of  the  condensers.  This  fact  may  be  readily  shown  to  follow 
from  the  as'sumption  that  the  amount  of  electrical  work  done 
when  the  machine  is  running  uniformly  is  directly  proportional  to 
the  time,  and  independent  of  the  capacity  of  the  condenser  used. 

On  account  of  the  uncertainty  of  the  conditions,  it  will  be 
necessary  to  take  a  large  number  of  observations,  and  to  use 
their  mean  in  testing  the  truth  of  the  above  statement. 

II. 

An  electrical  machine  is  a  generator  of  electricity,  and 
under  conditions  that  are  not  variable  it  has  a  constant 
electromotive  force,  and  a  constant  internal  resistance. 

If  the  machine  is  in  good  working  condition,  and  is  rotated 
uniformly  in  an  atmosphere  of  constant  humidity,  its  electro- 
motive force  and  internal  resistance  will  not  be  greatly  variable, 
provided  that  the  external  resistance  is  not  greater  than  a  few 
million  ohms. 

To  verify  this  statement,  connect  the  terminals  of  the 
machine  to  the  terminals  of  a  high-resistance  sensitive  galva- 
nometer. Turn  the  machine  uniformly,  and  observe  the  galva- 


136  JUNIOR   COURSE   IN    GENERAL   PHYSICS. 

nometer  readings  for  current,  both  direct  and  reversed.  Place 
in  series  with  the  machine  and  galvanometer  a  resistance  of 
100,000  ohms.  The  change  in  the  galvanometer  deflection  will 
probably  be  imperceptible.  This  shows  that  the  internal  resist- 
ance of  the  machine  is  enormous,  and  that  the  electromotive 
force  is  probably  very  great. 

In  order  to  determine  these  constants,  the  method  of  Exp. 
S2  may  be  used.  Place  in  series  with  the  galvanometer  and 
machine  a  variable  resistance  of  several  megohms.*  Observe 
the  galvanometer  readings  for  several  different  resistances  pro- 
ducing quite  different  deflections.  For  each  deflection,  the 
current  in  amperes  may  be  calculated  if  the  constant  of  the 
galvanometer  is  known. 
As  in  Exp.  S2,  we  have 

E 


If  E  and  RQ  are  constant,  they  may  be  determined  from  any 
pair  of  observations  giving  two  values  of  R  and  two  of  /. 
If  several  different  values  of  R  are  used,  plat  a  curve  with 
resistances  as  abscissas,  and  reciprocals  of  currents  as  ordinates. 
This  will  be  a  straight  line  from  whose  constants  E  and  R§ 
may  be  determined. 

Owing  to  the  difficulty  of  maintaining  a  uniform  rate  of 
rotation,  and  other  unavoidable  causes  affecting  the  current, 
the  values  of  the  current  used  in  computations  should  be  the 
mean  of  several  observations. 

It  must  be  remembered  that  the  potential  difference  of  the 
terminals  of  the  machine,  and  consequently  the  electromotive 
force,  is  enormously  greater  when  the  terminals  are  separated  by 
several  centimeters,  and  the  resistance  between  them  is  thou- 
sands of  megohms. 


*  Heavy  black-lead  lines  on  wood  will  serve  for  high  resistances.  If  divided  into 
sections  of  about  a  million  ohms,  these  resistances  may  be  measured  by  means  of  a 
Wheatstone  bridge. 


STATIC   ELECTRICITY.  137 

III. 

Replace  the  Ley  den  jars  by  large  ones.  Separate  the 
terminals  to  a  distance  of  8  or  10  cm.,  and  run  the  machine 
until  the  jars  are  charged.  Then  slip  off  the  belt,  and  stop 
the  revolving  plate  with  the  finger.  Under  favorable  circum- 
stances, the  plate  will  start  to  rotate  backwards,  and  continue 
to  do  so  for  quite  a  number  of  turns.  After  a  successful  trial, 
it  will  be  found  that  the  jars  are  very  nearly  discharged  when 
the  plate  ceases  to  rotate. 

Addenda  to  the  report: 

(1)  If  the  electrostatic  capacity  of   the  condenser  used  is 
known,   as    well   as   the   electrostatic    difference   of    potential 
producing  the  sparks  of  known  length,  calculate  the  electrical 
work  done  in  ergs  per  second,  in  watts. 

(2)  Prove    upon    theoretical    grounds    that   the   number  of 
sparks    per  minute   is   inversely  proportional   to   the  capacity 
of  the  condenser  used. 

(3)  Indicate  the  kind  of  battery  that  would  produce  an  effect 
similar  to  that  of   the    machine  in  the  second  experiment  of 
this  section. 

(4)  Indicate  the  cause  of  the  backward  rotation  in  the  third 
experiment  above. 


CHAPTER   V. 
GROUP  Q :   MAGNETISM. 

(Q)  General  statements ;  (QJ  Lines  of  force  and  the  study  of 
the  magnetic  field ;  (Q2)  Determination  of  the  magnetic 
moment  of  a  bar  magnet  by  the  method  of  oscillations ; 
(Q3)  Determination  of  magnetic  moment  by  the  magnetom- 
eter;  (Q4)  Measurement  of  the  intensity  of  a  magnetic  field ; 
(Q5)  Distribution  of  free  magnetism  in  a  permanent  magnet. 

(Q).    General  statements  concerning  magnetism. 

The  phenomena  of  current  electricity  and  of  magnetism  are 
almost,  if  not  quite,  inseparably  connected.  In  the  medium 
surrounding  a  conductor  conveying  a  current  of  electricity, 
magnets  are  acted  upon  by  a  force.  Such  a  region  is  naturally 
called  a  magnetic  field  of  force. 

Imaginary  lines  showing  at  all  points  the  direction  in  which 
the  force  acts  are  called  lines  of  force.  Greater  intensity  of  a 
field  of  force  is  usually  represented  by  a  greater  number  of  these 
lines  intersecting  a  given  area.  If  masses  of  iron  are  brought 
into  such  a  magnetic  field  of  force,  the  intensity  of  the  field  is 
greatly  increased,  in  the  neighborhood  of  those  parts  of  the 
iron  where  the  lines  enter  and  emerge.  The  same  is  also  true 
of  some  other  substances.  This  fact  may  be  explained  by 
saying  that  these  substances  are  much  better  conductors  of  lines 
of  force  than  the  air  or  ether,  or  that  their  "permeability"  for 
lines  of  force  is  greater  than  the  permeability  of  the  air.  Such 
good  conductors  of  magnetic  lines  of  force  are  called  magnetic 
substances. 

Those  portions  of  the  magnetic  lines  which  lie  within  a  mag- 
netic substance  are  called  lines  of  magnetization.  A  magnetic 

138 


MAGNETISM. 


139 


substance  containing  these  lines  is  said  to  be  magnetized,  and 
is  called  a  magnet.  Some  magnetic  substances,  steel  for 
example,  may  be  removed  from  the  magnetic  field  where  they 
have  been  magnetized,  without  losing  their  magnetic  properties. 
The  magnetic  field  surrounding  the  magnet  moves  with  the 
magnet,  and  seems  to  have  a  fixed  connection  with  it,  indepen- 
dent of  any  other  magnetic  field.  Such  a  body  is  called  a 
permanent  magnet. 

In  all  localities  where  the  experiment  has  been  performed, 
such  a  magnet  is  acted  on  by  a  force.  It  follows  that  there 
is  a  magnetic  field  surrounding  the  earth.  A  magnet  suspended 
in  a  magnetic  field  so  as  to  turn  freely  about  its  center  of 
gravity  always  comes  to  rest  with  its  longer  axis  in  a  particular 
direction.  The  direction  of  this  axis  is  always  tangent  to  lines 
of  force,  the  positive  direction  of  the  line  being  the  direction  in 
which  the  end  of  the  magnet  points,  which  points  north  in  the 
earth's  field.  The  end  of  a  magnet  which  points  north  in  the 
earth's  field  is  called  the  positive  end,  the  other  end  being  called 
negative. 

If  such  a  suspended  magnet  be  brought  into  a  field  about 
the  negative  end  of  another  magnet,  it  will  set  itself  with  its 
positive  end  pointing  towards  the  negative  end  of  the  other 
magnet.  The  reverse  is  true  in  the  field  about  the  positive  end 
of  the  second  magnet.  It  follows  from  this  that  the  lines  of 
force  of  the  field  due  to  a  magnet  diverge  from  its  positive  end, 
and  converge  towards  its  negative  end.  Such  a  region  within 
a  magnet,  towards  which  the  lines  of  force  converge,  or  from 
which  they  diverge,  is  called  a  pole. 

A  convenient  statement  of  the  fact  that  a  magnet  always 
tends  to  point  in  a  particular  direction  in  a  magnetic  field  may 
be  based  upon  the  principle  just  laid  down ;  viz.  : 

The  positive  pole  of  a  magnet  always  tends  to  move  along 
magnetic  lines  of  force  in  the  positive  direction,  and  the  nega- 
tive pole  in  the  negative  direction.  The  mutual  action 
two  magnets  when  brought  near  together  may  also  be 


140  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

in  the  following  form  :  Like  poles  repel  cadi  oilier,  and  unlike 
poles  attract  each  other. 

In  a  real  magnet,  lines  of  force  diverge  from  a  region  in  the 
positive  half,  curve  around  through  space,  and  converge  to  a 
region  in  the  negative  half,  and  then  pass  on  through  the  mag- 
net as  lines  of  magnetization.  The  idea  of  a  pole,  as  a  point 
towards  which  lines  of  force  converge,  is  a  highly  idealized 
conception.  It  is  a  very  useful  conception,  however,  and  by 
most  authors  is  made  the  basis  of  the  whole  system  of  electro- 
magnetic units.  This  ideal  conception  of  a  magnet  pole  is 
not  likely  to  lead  to  error  except  in  one  case ;  namely,  when 
the  intensity  of  the  force  at  a  point  in  a  magnetic  field  is 
expressed  as  a  function  of  the  strength  of  its  poles,  and  the 
distance  between  them,  as  in  Exp.  O3. 

An  ideal  magnet  with  ideal  poles  of  a  given  strength  and 
a  given  distance  between  them  may  be  conceived,  such  that  the 
magnetic  field  would  be  at  four  symmetrical  points,  exactly 
like  the  field  produced  by  a  real  magnet.  But  the  field  of  the 
real  magnet  would  be  different  at  all  other  points  from  the  field 
of  the  ideal  magnet.  For  example,  the  field  of  a  magnet,  quite 
close  to  its  middle  point,  is  such  as  would  be  produced  by  an 
ideal  magnet  with  poles  comparatively  close  together,  while 
the  reverse  is  true  for  a  point  near  either  end  of  the  magnet. 
The  error  introduced  into  Exp.  Q3  by  the  assumption  made 
is  quite  small  whenever  L  >  3  /. 

Notwithstanding  what  has  been  said  about  magnet  poles, 
the  term  "magnetic  moment "  has  a  perfectly  definite  physical 
meaning.  If  a  magnet  be  placed  in  a  magnetic  field  with  its 
axis  at  right  angles  to  the  lines  of  force,  it  will  be  acted  upon 
by  a  turning  force.  If  the  moment  of  this  turning  force  be 
represented  by  G,  and  the  intensity  of  the  field  by  H,  the 
magnetic  moment  of  the  magnet  may  be  defined  by  the  re,la- 

tion 

MH=G,  (101) 

in  which  M  is  the  magnetic  moment. 


MAGNETISM. 


141 


EXPERIMENT  Qr  Lines  of  force  and  the  study  of  magnetic 
fields. 

Surrounding  every  magnet  and  every  current  of  electricity 
there  is  a  magnetic  field.  The  earth  also  has  a  magnetic  field 
surrounding  it.  The  object  of  this  experiment  is  to  investigate 
the  direction  in  which  the  force  acts  in  such  fields;  that  is 
to  say,  the  direction  of  the  lines  of  force. 

For  this  purpose  place  a  sheet  of  glass  immediately  above 
the  magnet  whose  field  is  to  be  investigated,  and  scatter  over  it 
iron  filings,  allowing  them  to  drop  from  a  height  of  8  or  10  inches. 
If  the  magnet  is  sufficiently  strong,  the  filings  will  arrange  them- 
selves in  "  lines  offeree."  A  slight  tapping  or  jarring  of  the 
glass  will  probably  make  the  magnetic  curves  more  perfect. 
Sheet  metal  (not  iron)  or  paper  may  be  used  instead  of  glass  if 
desired,  but  the  glass  plate  has  the  advantage  of  allowing  the 
position  of  the  magnet  to  be  clearly  seen.  Permanent  records 
of  the  curves  may  be  obtained  by  allowing  the  filings  to  arrange 
themselves  upon  a  sheet  of  blue  print  paper,  and  exposing  the 
latter  to  the  sun  while  the  filings  are  still  in  position. 

Among  cases  which  may  be  studied  to  advantage  in  this 
manner  are  the  following : 

(1)  The  field  of  a  single  "  horseshoe  "  magnet. 

(2)  Two  magnets  with  like  poles  near  each  other. 

(3)  Two  magnets  with  unlike  poles  near  each  other. 

(4)  A  bar  magnet  placed  in  the  neighborhood  of  a  horse- 
shoe magnet. 

(5)  The  field  of   two  horseshoe  magnets  placed  vertically, 
their  four  poles  forming  a  square. 

Many  other  more  complicated  arrangements  will  suggest 
themselves.  Observe  also  the  effect  of  pieces  of  soft  iron, 
placed  in  different  positions  in  the  field,  upon  the  form  of  the 
curves  obtained.  If  the  piece  of  soft  iron  seems  to  produce 
little  effect,  bring  it  in  contact  with  one  pole. 

The  direction  of  the  magnetic  force  at  any  point  will  be 
indicated  by  the  direction  in  which  a  small  compass  needle 


142  JUNIOR   COURSE    IN   GENERAL   PHYSICS. 

will  set  itself  when  placed  at  that  point.  By  shifting  the 
compass  from  place  to  place,  the  direction  of  the  force  can 
thus  be  found  at  any  number  of  points. 

To  study  the  field  by  this  method,  place  one  or  more 
magnets  in  the  middle  of  a  large  board  ruled  in  squares,  which 
has  been  previously  set  with  two  opposite  edges  parallel  to 
the  magnetic  meridian.  The  board  should  be  so  large  that 
at  the  edges  the  field  due  to  the  magnets  is  decidedly  weaker 
than  the  earth's  "field. 

By  means  of  a  small  compass  determine  the  direction  of 
the  lines  of  force  for  a  large  number  of  points.  There  should 
be  enough  of  these  observations,  so  that  the  direction  in 
which  the  compass  needle  would  point  if  placed  anywhere  on 
the  board  may  be  known  within  rather  narrow  limits. 

Make  a  diagram  (to  scale)  of  the  board  and  magnets,  and 
at  each  point  where  the  compass  was  placed  draw  a  little  arrow 
to  show  the  direction  of  the  force.  An  arrow  should  also  be 
drawn  somewhere  on  the  board  to  show  the  direction  of  the 
earth's  field.  Map  the  whole  field  on  the  board  by  means 
of  lines  representing  the  lines  of  force.  These  lines  do  not 
need  to  pass  through  the  arrows,  but  shquld  be  so  drawn 
as  to  represent  the  direction  in  which  the  compass  needle 
would  point  if  placed  upon  corresponding  points  on  the 
board. 

The  field  so  mapped  is  the  resultant  field  of  the  magnets 
and  of  the  horizontal  component  of  the  earth's  field.  Therefore, 
it  must  not  be  expected  that  all  the  lines  of  force  will  enter 
a  magnet. 

In  the  neighborhood  of  every  magnet  or  system  of  magnets 
there  are  in  general  two  or  more  points  where  the  magnetic 
field  due  to  them  exactly  neutralizes  the  earth's  field.  At 
these  points  there  will  be  no  directive  force  acting  on  the 
compass  needle,  and  on  opposite  sides  of  these  points  the 
needle  will  point  in  opposite  directions.  Locate  these  points 
on  your  diagram. 


MAGNETISM. 


143 


Addenda  to  the  report: 

(1)  State  the  law  of  magnetic  force. 

(2)  Define :    unit   magnet    pole ;    magnetic   field   of   force ; 
field    of    unit    intensity ;     magnetic    difference    of    potential ; 
magnetic  potential   at   a   point ;    equipotential  surfaces ;    mag- 
netic   lines    of    force ;    unit    line   of    force,    or   unit    tube   of 
force. 

(3)  Show   that   lines    of   force   and    equipotential    surfaces- 
are  mutually  perpendicular. 

(4)  Indicate  the  reason  why,  in  triis  experiment,  the  filings 
move  away  from  points  directly  above  the  magnet,  especially 
in  the  neighborhood  of  the  poles. 

(5)  Draw  the  horizontal   sections  of  several    equipotential 
surfaces  whose  potentials  differ  by  equal  amounts. 

(6)  Assume  that  the  potential  of  any  point   a    centimeter 
from  the  north  pole  is  100,  and  that  the  surface  of  zero  potential 
bisects  the  distance  between  the  poles,  and  determine  approxi- 
mately from  the  map  and  from  the  assumptions  already  made, 
the   magnetic   force   at-  several   points    10   or    20   cm.    distant 
from  the  magnet. 

EXPERIMENT  Q2.  Determination  of  the  magnetic  moment 
of  a  bar  magnet  by  the  method  of  oscillations. 

A  magnet  suspended  by  a  torsionless  fiber  with  its  axis 
horizontal  will  come  to  rest  with  its  magnetic  axis  in  the  mag- 
netic meridian.  If  the  magnet  is  turned  so  as  to  make  a  small 
angle  with  the  magnetic  meridian,  the  moment  of  the  force 
tending  to  restore  the  magnet  to  its  position  of  equilibrium  is 
directly  proportional  to  the  angular  displacement.  The  result- 
ing motion  of  the  magnet,  when  left  free  to  vibrate,  is  therefore 
a  simple  harmonic  motion. 

The  periodic  time  is  dependent  upon  the  magnetic  moment  of 
the  magnet,  its  moment  of  inertia,  and  the  horizontal  intensity 
of  the  magnetic  field  in  which  it  is  suspended.  The  following 
equation  may  be  derived  by  equating  the  kinetic  energy  of  the 


144  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

magnet  at  its  mid-position  to  the  potential  energy  when  at 
the  end  of  its  swing.  The  method  of  derivation  is  the  same 
as  that  pursued  in  Exp.  Er 

(102) 


To  perform  the  experiment  : 

(1)  Place  the  magnet  in  a  small  wire  stirrup,  and  suspend  it 
by  a  few  untwisted  silk  fibers.     It  should  be  suspended  in  a  box 
with   glass   ends,    to  avoid  the  effect  of   air  currents,    and    a 
position  should  be  chosen  at  a  distance  from  movable  masses  of 
iron.     If  the  bar  is  rather  strongly  magnetized,  the  torsion  of 
the  silk  fiber  may  be  neglected,  or,  if  desired,  it  may  be  elimi- 
nated by  determining  the  ratio  of  the  moment  of  torsion  to  the 
moment  of  the  magnetic  forces.* 

(2)  Set  the  magnet  to  vibrating  through  an  arc  of  not  more 
than  five  degrees.      Determine  the  period  of  oscillation  by  the 
method  of  Exp.  A6,  or  by  counting  the  number  of  passages  in 
the  same   direction   during   five  or  six  minutes.      The   period 
should    be    determined    several    times,    and    as    carefully    as 
possible. 

(3)  Measure  the  length,  diameter,  and  mass  of  the  bar,  and 
from  these  data  compute  its  moment  of  inertia.     If  the  value  of 
the  horizontal  component  of  magnetism  for  the  place  where  the 
magnet  was  suspended  is  known,  the  value  of  M  may  be  com- 
puted from  the  above  equation. 

The  method  of  oscillations  may  be  used,  if  desired,  to  deter- 
mine the  value  of  H  in  different  parts  of  the  laboratory  ;  in 
which  case  the  period  of  oscillation  must  first  be  determined 
from  observations  taken  in  a  locality  where  H  is  known. 

If  this  experiment  and  the  following  one  be  performed  with 
the  same  magnet  and  at  the  same  place,  both  H  and  M  may  be 
determined  in  absolute  measure. 

*  Kohlrausch,  Physical  Measurements,  p.  128. 


MAGNETISM.  !45 

Addenda  to  the  report: 

(1)  State  the  effect  upon  the  period  of  oscillation,  if  in  the 
above  experiment  the  magnet  were  not  quite  horizontal. 

(2)  State  the  effect  upon  the  period,  if  the  magnet  were 
bent  into  the  form  of  a  horseshoe,  without  changing  its  intensity 
of  magnetization. 

(3)  Determine  the  average  intensity  of  magnetization  in  the 
magnet  experimented  with,  and  indicate  in  what    part  of   the 
magnet  it  is  the  greatest,  and  in  what  part  the  least. 

(4)  Compute  the  ergs  of  work  that  would    be  required  to 
rotate  the  magnet  180°  about  a  vertical  axis  in  the  earth's  field 
from  the  position  of  rest. 

(5)  Assuming  the  dip  75°,  compute  the  work  required  to 
rotate  the  magnet  from  a  vertical  position  through  180°  about  a 
horizontal  axis. 

EXPERIMENT  Q3.  Determination  of  magnetic  moment  by  the 
magnetometer. 

When  two  forces  act  at  right   angles  to  each  other,  their 
resultant  makes  an  angle  with  each  of  the  forces  such  that  the 
tangent  is  the   ratio  of   the   two  forces.      See 
Fig.  46,  in  which  a  and  b  are  the  forces  and  « 
and  /3  the  angles  which  their  resultant  r  makes 
with  them  respectively. 

Obviously,  tana=-  and  tan/3  =  -- 
a  b 

This  fact  may  be  used  to  determine  the  ratio 
of  the  intensity  of  two  magnetic  fields  at  any 
given  point. 

When  a  magnet  is  so  placed  that  the  field  due  to  it  at  a 
given  point  is  at  right  angles  to  the  field  due  to  the  earth,  a 
short  magnetic  needle  placed  at  that  point  will  be  deflected 
from  the  magnetic  meridian  through  an  angle  whose  tangent  is 
equal  to  the  ratio  between  the  intensities  of  the  two  components 
of  the  field  at  that  point. 

VOL.  I  —  L 


146  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

The  needle  must  be  short  with  respect  to  the  distance  to 
the  magnet,  for  otherwise  its  ends  would  extend  too  far  beyond 
the  point  at  which  the  field  of  the  magnet  has  the  intensity 
given  below. 

The  strength  of  the  field  at  any  point  due  to  a  magnet  is  a 
known  function  of  its  magnetic  moment,  the  distance  between 
its  poles,  and  the  distance  of  the  point  from  the  magnet. 


HB 


I 


Fig.  47. 


For  example,  the  strength  of  the  field  at  A,  due  to  a  magnet 
whose  poles  are  at  n  and  s  (Fig.  47),  is 


2  ML  (      . 


At  B,  the  strength  of  the  field  due  to  the  same  magnet  is 

(104) 


in  which  M  is  the  magnetic  moment  of  the  magnet,  2.  1  the  dis- 
tance between  its  poles,  and  L  the  distance  of  A  or  B  from  the 
center  of  the  magnet. 

If  the  magnet  is  at  right  angles  to  the  magnetic  meridian,  a 
magnetic  needle  placed  at  A  will  be  deflected  from  the  meridian 
through  an  angle  B  according  to  the  relation 

=<tanS,  (,0S) 


and  there  will  be  a  corresponding  relation  for  the  position  B. 


MAGNETISM. 


147 


These  equations  may  also  be  derived  by  an  application  of 
the  principle  of  moments  to  the  magnetic  forces  acting  on  the 
suspended  needle. 

The  magnetometer  consists,  essentially,  of  a  suspended 
magnetic  needle,  a  bar,  I  m.  or  more  long,  upon  which  to  place 
the  magnet  with  which  the  experiment  is  to  be  performed,  and 
some  means  of  measuring  the  angle  through  which  the  needle 
is  turned. 

In  preparation  for  this  experiment,  adjust  the  magnetometer 
bar  at  right  angles  to  the  magnetic  meridian.  This  may  be 
done  with  sufficient  accuracy  with  the  aid  of  a  small  compass 
needle.  If  greater  accuracy  is  desired,  adjust  the  magnetom- 
eter bar  by  trial  so  that  the  same  deflection  is  produced  when 
the  magnet  is  placed  on  opposite  sides  of  the  magnetometer 
needle,  at  the  same  distance  from  it,  and  with  the  same  pole 
pointing  towards  it. 

Having  completed  the  adjustment,  proceed  as  follows: 

(1)  Place  the  magnet   on  the  bar  with  its  poles  pointing 
east   and   west,    and    at    a   distance    of    not   less    than    20    or 
30  cm.  east  of  the  magnetometer  needle. 

(2)  Observe    the    magnetometer    reading   by    means   of    a 
telescope  and   scale,  and  the  mirror  on  the  magnetic  needle. 
Then  turn  the  magnet  end  for  end,  keeping  its  distance  from 
the  needle  the  same,  and  again  observe  the  reading.     Half  the 
difference  of  the  two  readings  is  a  measure  of  the  deflection 
of    the   needle   from    its    position    of   rest    on  account   of   the 
presence  of  the  magnet. 

(3)  Reverse  the  magnet  in  this  way  several  times  so  as  to 
get  the  average  of  a  number  of  observations. 

(4)  Finally  place  the  magnet  at  the  same  distance  to  the  west 
of  the  magnetometer  needle  and  proceed  as  before.     From  the 
average  of  all  the  deflections  observed,  and  the  distance  between 
the  mirror  and   scale,   compute  the  angle  through  which  the 
needle  is  deflected. 

As  a  check  the  observations    should   be  repeated  with  the 


148 


JUNIOR   COURSE   IN    GENERAL   PHYSICS. 


magnet  at  such  a  distance  from  the  needle  as  to  produce  a 
deflection  which  is  considerably  greater  or  less  than  that  first 
used. 

The  following  table  gives  typical  data  and  shows  the  method 
of  presenting  them. 

MAGNETIC  MOMENT  BY  THE  MAGNETOMETER. 


North  Pole 
Pointing. 

Distance  of 
Center  of  Mag. 
from  Needle. 

Scale 
Reading. 

Deflection 
in  Scale  Div. 

Distance  between  poles 
of  magnet,                2  / 
Horizontal  intensity 
Average  deflection 

=  22  cm. 

=    0-H5 
-    45-91 

48.37 

Distance  from  mirror  to 

West 

54  W. 

3-04 

45-33 

scale           =91.1  scale  division 

East 

54  W. 

94.62 

46.25 

tan  26 

=  0-5037 

East 

54  E. 

95-63 

47-37 

26 

=  26°  44' 

West 

54  E. 

3-57 

44.69 

tan0 

=  0.2376 

48.26 

Magnetic  moment 

=     2436 

West 
East 
East 

80  W. 
80  W. 
80  E. 

35-86 
60.68 
60.85 

12.40 
12.42 
12.59 

Average  deflection 
tan  2  0' 

26' 

=    12.43 
=  0.1364 
=  7°  46' 

West 

80  E. 

35-95 

12.31 

tanfl' 

=  0.0679 

48.26 

Magnetic  moment 

=     2426 

In  the  above  formula,  2  /  is  the  distance  between  the  poles 
of  the  magnet,  and  is  therefore  less  than  the  length  of  the  bar 
itself.  The  position  of  the  poles,  and  therefore  the  length  2  /, 
may  be  approximately  determined  by  the  aid  of  a  small  compass. 

If  H  is  known,  M  may  be  computed  ;  or,  if  the  product  MH 
is  known,  both  M  and  H  may  be  computed  in  absolute  measure. 
This  product  may  be  obtained  by  the  method  described  in 
Exp.  Qa. 

If  the  magnetometer  admits  of  it,  a  similar  series  of  obser- 
vations should  be  taken  with  the  magnet  placed  at  points  north 
and  south  of  the  magnetometer  needle,  its  poles  pointing  east 
and  west  as  before. 

Addenda  to  the  report: 

(i)  Determine  the  strength  of  each  pole  of  the  magnet 
experimented  with. 


MAGNETISM. 

(2)  Calculate  the  magnetic  force  and  potential  due  to  the 
magnet  for  two  or  three  points  in  its  neighborhood. 

(3)  Calculate  the  work  required  to  carry  a  magnet  pole  of 
strength  equal  to  either  pole  of  the  magnet,  from  one  pole  face 
to  the  other  pole  face,  along  any  path. 

(4)  Compare  the  pull  on  either  magnetic  pole  with  the  pull 
of  gravity  on  one  gram  for  a  case  in  which  the  inclination  ot 
the  earth's  lines  of  force  is  75°. 

EXPERIMENT  Q4.  Measurement  of  the  intensity  of  a  mag- 
netic field. 

The  intensity  of  a  magnetic  field  at  different  points  may  be 
compared  with  the  intensity  of  the  earth's  field  by  either  of  the 
methods  made  use  of  in  Exps.  Q2  and  Q3.  In  the  following 
experiment,  these  methods  are  to  be  used  in  measuring  the 
magnetic  field  at  a  series  of  points  in  the  neighborhood  of  a 
permanent  magnet. 

I. 

Place  the  magnet  with  its  axis  in  the  magnetic  meridian, 
its  negative  pole  pointing  north.  For  all  points  to  the  east  or 
west  of  the  middle  of  the  magnet,  the  intensity  of  the  field  will 
be  the  arithmetical  sum  of  the  earth's  horizontal  intensity  and 
the  intensity  of  the  field  at  that  point,  due  to  the  magnet. 
For  all  points  to  the  north  or  south  of  the  magnet,  the  inten- 
sity of  the  field  will  be  the  difference  of  these  two  quan- 
tities. 

Determine  the  period  of  oscillation  of  a  small  magnet  of  any 
shape,  for  a  series  of  points  on  a  line  at  right  angles  to  it,  and 
bisecting  the  distance  between  its  poles.  Do  the  same  for  a 
series  of  points  north  or  south  of  the  magnet.  For  each  point, 
the  number  of  oscillations  produced  in  three  or  four  minutes 
should  be  determined. 

As  the  intensity  of  the  field  due  to  the  magnet  varies  most 
rapidly  near  it,  the  points  of  observation  should  be  closer 


150  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

together  the  nearer  they  are  to  the  magnet.      A  good  series 
of  distances  is  the  geometric   series  -£$L,  ^L,  \L,  •••  2L,  in 
which  L  is  the  length  of  the  magnet. 
As  in  Exp.  Q2,  we  have 

(I06) 


in  which  HP  is  the  intensity  of  the  field  due  to  the  magnet  at 
the  point  where  the  time  of  vibration  is  TP)  H  is  the  horizontal 
intensity  of  the  earth's  magnetism,  and  C  is  a  constant  depend- 
ing upon  the  magnet.  C  may  be  eliminated  by  taking  the  time 
of  vibration  at  a  distance  from  the  magnet  where  HP  is  zero. 
Hp  can  then  be  computed  in  terms  of  H,  or  if  H  is  known,  it 
may  be  computed  in  absolute  measure. 

Plat  a  curve  with  distances  from  the  magnet  as  abscissas, 
and  corresponding  values  of  HP  as  ordinates. 

II. 

Place  a  large  bar  magnet  at  right  angles  to  the  mag- 
netic meridian,  as  in  Exp.  Q3.  For  points  "A"  and  "B," 
the  ratio  of  the  intensity  of  the  field,  due  to  the  magnet  and 
the  earth's  horizontal  intensity,  will  be  equal  to  the  tangent 
of  the  angle  through  which  a  magnetic  needle  will  be  deflected 
from  the  magnetic  meridian,  if  placed  at  that  point. 

Place  a  compass  with  a  circle  graduated  to  degrees  at  a 
series  of  points  "A,"  at  distances  from  the  magnet  as  in  I 
above.  For  each  point  determine  the  deflection  from  the 
meridian  as  follows :  Read  both  ends  of  the  needle,  then 
reverse  the  magnet  and  read  again.  The  angle  through  which 
the  needle  has  been  turned  is  double  the  angle  of  deflection 
from  the  meridian.  In  the  same  manner  make  a  series  of 
observations  for  points  "B."  Plat  a  curve  with  distances  from 
the  magnet  as  abscissas,  and  the  intensity  of  the  field  due  to 
the  magnet  as  ordinates. 


MAGNETISM.  151 

Addenda  to  the  report: 

(1)  From    the  equations  in    Exp.    Q3,   and    several    points 
on    one   of   the   above    curves,    compute   values   of    M.      The 
points  taken  should  not  be  observed  points  unless  these  points 
happen  to  fall  exactly  upon  the  curve. 

(2)  Note  whether  these  values  of  M  show  a  progressive 
increase  or  decrease,  and  if  so,  indicate  cause  of  the  variation. 

EXPERIMENT  Q5.  Distribution  of  "free"  magnetism  in 
a  permanent  magnet. 

For  purposes  of  calculation,  the  distribution  of  imaginary 
magnetic  matter  in  a  magnet  may  be  considered  in  two  ways : 
as  a  distribution  throughout  its  volume,  or  over  its  surface. 
The  volume  distribution  or  intensity  of  magnetization  is  great- 
est midway  between  the  poles.  The  surface  distribution 
is  greatest  near  the  ends,  and  is  vanishingly  small  mid- 
way between  the  poles.  This  imaginary  magnetic  matter  is 
supposed  to  be  so  distributed  as  to  produce  by  its  attraction 
or  repulsion  the  same  field  of  force  that  the  magnet  produces. 
From  this  we  see  that  the  quantity  of  surface  magnetism,  or 
"free"  magnetism,  as  it  is  called,  is  everywhere  proportional 
to  the  number  of  unit  lines  of  force  which  enter  or  emerge 
from  the  magnet. 

I. 

The  distribution  of  magnetism  may  be  determined  by 
measuring  the  force  necessary  to  detach  a  small,  soft  iron 
armature  from  the  magnet.  For  measuring  this  force  use  a 
pair  of  balances  or  a  spiral  spring,  whose  extension  can  be 
readily  determined. 

Determine  in  this  way  the  force  necessary  to  detach  the 
armature  for  ten  or  twenty  points  along  the  magnet  from  one 
end  to  the  other.  The  magnet  may  not  be  symmetrically 
magnetized  ;  if  not,  the  forces  at  symmetrical  points  will  not 
be  equal.  Plat  a  curve  with  distances  from  the  center  of  the 


!52  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

magnet  as  abscissas,  and  the  forces   necessary  to  detach  the 
armature  as  ordinates. 

In  considering  that  this  curve  represents  the  distribu- 
tion of  magnetism  along  the  magnet,  two  things  should  be 
remembered  : 

(1)  By  induction,  the  distribution  of  magnetism  is  slightly 
changed  on  account  of  the  presence  of  the  armature.     If  the 
armature  is  quite  small  with  respect  to  the  magnet,  this  may 
be  neglected. 

(2)  The  force  with  which  the  soft  iron  armature  is  attracted 
to  the  magnet  is  proportional  to  the  square  of  the  magnetism 
at  that  point.     This  is  true,  since  the  force  is  proportional  to 
the  product  of  the  magnetism  of  the  magnet  and  the  magnetism 
of  the  armature,  in  the  immediate  neighborhood  of  the  point 
of  contact,  but  the  induced  magnetism  of  the  armature  is  itself 
proportional  to  the  magnetism  of  the  permanent  magnet. 

II. 

The  distribution  of  magnetism  may  also  be  determined 
by  the  method  of  oscillations. 

Place  a  bar  magnet  in  a  vertical  position,  and  determine 
the  period  of  oscillation  of  a  small  magnet  for  a  series  of 
positions  along  the  magnet,  and  quite  close  to  it.  These 
points  should  be  north  or  south  of  the  magnet. 

As  in  experiment  Q4,  we  have 

(107) 


in  which  HP  is  the  intensity  for  the  point  P  of  the  field  due 
to  the  magnet,  resolved  in  a  direction  perpendicular  to  its 
length. 

If  the  point  P  is  quite  close  to  the  magnet,  HP  will 
be  proportional  to  the  free  magnetism  at  the  corresponding 
point  of  the  magnet.  As  in  (i),  plat  a  curve  with  distances 
from  the  center  of  the  magnet  as  abscissas,  and  corresponding 
values  of  HP  as  ordinates. 


CHAPTER   VI. 
GROUP  R:    THE  ELECTRIC  CURRENT. 

(R)  General  statements ;  (Rx)  The  law  of  the  tangent  galvanom- 
eter;  (R2)  Measurement  of  current  by  electrolysis ;  (R3) 
Measurement  of  the  constant  of  a  sensitive  galvanometer ; 
(R4)  Theory  of  shunts ;  (R5)  Measurement  of  current  by 
means  of  the  galvanometer. 

(R).   General  statements  concerning  the  electric  current. 

The  electric  current  may  be  denned  as  the  rate  at  which 
electrification  is  transferred,  or  the  amount  of  electricity  which 
passes  through  a  given  plane  cutting  the  circuit  at  right  angles 
to  the  lines  of  flow,  in  a  unit  time.  When  two  bodies  which 
differ  in  potential  are  connected  by  means  of  a  conductor,  the 
fleeting  phenomena  which  accompany  the  electric  discharge 
occur,  and  we  have  a  transient  current ;  if  the  difference  of 
potential  be  maintained  constant  by  the  expenditure  of  work, 
there  will  be  a  permanent  current. 

If  current  were  always  measured  by  electrolysis,  the  idea  of 
current  would  be  a  derived  conception  involving  time.  Since, 
however,  when  a  current  flows  in  a  conductor  there  is  a  magnetic 
field  surrounding  the  conductor,  the  intensity  of  which  at  any 
given  point  is  always  directly  proportional  to  the  current,  it  is 
more  convenient  to  measure  the  latter  by  means  of  the  field 
which  it  produces.  In  this  way  we  reach  a  conception  of  cur- 
rent which  does  not  directly  involve  time. 

Those  instruments  which  measure  currents  by  comparing 
the  field  produced  with  the  earth's  magnetic  field,  or  with  the 

153 


154 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


earth's  field  as  modified  by  controlling  or  regulating  magnets, 
are  called  galvanometers.. 

The  lines  of  force  surrounding  a  wire  carrying  a  current 
may  easily  be  mapped  by  the  aid  of  iron  filings.  Figs.  48  and 
49  are  such  maps  showing  the  field  around  a  straight  wire. 
The  former  was  obtained  by  cutting  a  sensitive  plate  and 
passing  the  conductor,  the  field  of  which  was  to  be  mapped, 


Fig.  48.  —  Map  of  the  Field  around  a  Wire  carrying 
Current  (from  a  Photograph). 


Fig.  49. 


through  the  hole.  The  plate  was  then  fastened  in  a  position  at 
right  angles  to  the  conductor.  Current  was  sent  through  the 
latter,  and  the  surface  of  the  film  was  strewn  with  iron  filings. 
These  operations  having  been  completed  by  the  red  light  of  the 
developing-room,  the  plate  was  then  exposed  for  three  seconds 
to  gas-light,  after  which  the  photograph  was  developed,  giving 
the  map. 

The  direction  of  the  lines  of  force,  at  any  point  in  the  field, 
produced  by  a  current,  is  always  at  right  angles  to  the  plane 
containing  the  point  and  the  current.  The  intensity  of  this 
field  may  be  deduced  from  Laplace's  law,  of  which  the  following 
equation  is  a  statement : 

/"cin    (Mr 

(108) 


/  sin  6ds 


THE   ELECTRIC   CURRENT. 

In  this  expression,  dHP  is  the  intensity  of  the  magnetic  field 
at  the  point  P,  due  to  the  short  element  ds  of  the  current  of 
intensity  /;  r  is  the  distance  from  the  point  P  to  the  element 
ds,  and  6  is  the  angle  which  the  direction  of  the  element  ds 
makes  with  the  line  drawn  from  it  to  the  point  P. 

The  absolute  unit  of  current  in  the  electromagnetic  system 
is  that  current,  a  unit  length  of  which  will  produce  unit  magnetic 
field  at  unit  distance  from  the  current.  It  follows  that  the 
C.  G.  S.  unit  current  in  the  electromagnetic  system,  flowing 
around  a  circle  of  i  cm.  radius,  will  produce  at  the  center  of  the 
circle  a  field  whose  intensity  is  2  TT  units. 

The  Chamber  of  Delegates  of  the  Electrical  Congress  at 
Chicago  adopted  "as  a  {practical}  unit  of  current  the  inter- 
national ampere,  which  is  one-tenth  of  the  unit  of  current  of  the 
C.  G.  S.  system  of  electromagnetic  units,  and  which  is  represented 
sufficiently  well  for  practical  tise  by  the  unvarying  current  which, 
when  passed  through  a  solution  of  nitrate  of  silver,  in  water,  and 
in  accordance  with  the  accompanying  specifications*  deposits  silver 
at  the  rate  of  o.ooi  1 18  grams  per  second" 


*  In  this  specification,  the  term  "  silver  voltameter  "  means  the  arrangement  of 
apparatus  by  means  of  which  an  electric  current  is  passed  through  a  solution  of  nitrate 
of  silver  in  water.  The  silver  voltameter  measures  the  total  electrical  quantity  which 
has  passed  during  the  time  of  the  experiment,  and  by  noting  this  time,  the  time-average 
of  the  current,  or  if  the  current  has  been  kept  constant,  the  current  itself  can  be 
deduced. 

In  employing  the  silver  voltameter  to  measure  currents  of  about  I  ampere,  the 
following  arrangements  should  be  adopted  : 

The  cathode  on  which  the  silver  is  to  be  deposited  should  take  the  form  of  a 
platinum  bowl,  not  less  than  10  cm.  in  diameter,  and  from  4  to  5  cm.  in  depth. 

The  anode  should  be  a  plate  of  pure  silver  some  30  sq.  cm.  in  area,  and  2  or 
3  mm.  in  thickness. 

This  is  supported  horizontally  in  the  liquid  near  the  top  of  the  solution  by  a 
platinum  wire  passed  through  holes  in  the  plate  at  opposite  corners.  To  prevent  the 
disintegrated  silver  which  is  formed  on  the  anode  from  falling  on  to  the  kathode,  the 
anode  should  be  wrapped  round  with  pure  filter  paper  secured  at  the  back  with 
sealing-wax. 

The  liquid  should  consist  of  a  neutral  solution  of  pure  silver-nitrate  containing 
about  15  parts  by  weight  of  the  nitrate  to  85  parts  of  water. 

The  resistance  of  the  voltameter  changes  somewhat  as  the  current  passes.     To 


56 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


The  intensity  of  the  field  at  the  center  of  the  galvanometer 
coils  produced  by  unit  current  flowing  in  the  coils  is  called  the 
true  constant  of  the  galvanometer,  and  is  generally  denoted  by 
G.  For  many  galvanometers,  this ,  constant  may  be  computed 
from  Laplace's  law,  and  the  dimensions,  position,  and  number 
of  turns  of  the  coils. 

If  a  galvanometer  coil  is  placed  with  the  plane  of  its 
windings  in  the  magnetic  meridian,  the  field  produced  by  it 
at  the  center  of  the  coil  (or  at  any  point  on  its 
axis)  will  be  at  right  angles  to  the  earth's  field. 
Let  CO  (Fig.  50)  represent  the  horizontal  section 
of  the  galvanometer  coil.  The  intensity  of  the 
field  at  the  point  <9,  due  to  the  current  /  flowing 
in  the  coils  is,  GL  If  H  represents  the  horizontal 
intensity  of  the  field  at  the  point  O  due  to  the 
earth  (and  regulating  magnets),  the  resultant  of 
these  two  fields  will  make  an  angle  8  with  the 
plane  of  the  coils  such  that 


Gl 


/  =  — tanS. 
G 


(109) 


Fig.  50. 


If  a  short  magnetic  needle  be  suspended  at  the 
point  O,  it  will  come  to  rest  with  its  magnetic  axis 
in  the  plane  of  the  resultant  magnetic  field  through 
the  point  O.  That  is,  it  will  turn  through  the 
angle  8  from  the  position  of  equilibrium  when  no  current  flows. 
If  the  magnetic  needle  is  not  short,  its  ends  are  liable  to  extend 
too  far  beyond  the  point  O  at  which  the  field  due  to  the  current 
7  has  the  intensity  GL  If  the  current  is  required  in  amperes 
instead  of  in  absolute  units  of  current,  the  above  equation  becomes 

7  =  io~tanS.  (no) 


prevent  these  changes  having  too  great  an  effect  on  the  current,  some  resistance 
besides  that  of  the  voltameter  should  be  inserted  in  the  circuit.  The  total  metallic 
resistance  of  the  circuit  should  not  be  less  than  10  ohms. —  [Extract  from  Bulletin  30 
U.  S.  Coast  and  Geodetic  Survey,  embodying  the  specifications  referred  to  above.] 


THE   ELECTRIC   CURRENT. 


157 


TT 

The  constant  quantity  10—    is  called  the   "reduction  factor," 

G 

the   "working  constant,"  or,  for  brevity,  simply  the  constant 
of  the  galvanometer.     If  this  be  represented  by  /0,  we  have 

7  =  /0tanS.  (ill) 

From  (in)  it  is  obvious  that  the  galvanometer  constant  70  is 
that  current  which  will  produce  a  deflection  of  45°. 

In  this  discussion  it  is  assumed  that  the  friction  of  the 
needle  on  the  pivot  or  the  torsion  of  the  suspending  fiber  is 
negligible.  This  is  generally  a  safe  assumption  except  in 
sensitive  galvanometers,  where  a  very  small  needle  or  an  astatic 
system  is  used.  In  such  cases,  the  moment  of  the  force  of 
torsion  tending  to  bring  the  needle  back  to  its  position  of 
equilibrium  may  be  very  considerable  compared  with  the 
moment  of  the  magnetic  forces  tending  to  return  the  needle 
to  the  magnetic  meridian.  Moreover,  there  may  be  a  twist  in 
the  fiber,  such  that  the  needle  does  not  return  to  the  magnetic 
meridian  when  the  current  ceases  to  flow  in  the  galvanometer 
coils. 

In  Fig.  50,  above,  it  is  obvious  that  if  the  galvanometer 
current  is  reversed,  the  direction  of  the  field  GI  will  be 
reversed.  In  this  case  the  magnetic  needle  will  be  turned 
through  the  same  angle  S,  with  its  north  end  pointing  on  the 
opposite  side  of  the  meridian.  This  offers  a  means  of  setting 
the  galvanometer  coils  in  the  plane  of  the  magnetic  meridian,  if 
they  are  not  already  so  adjusted. 

For  this  purpose,  we  send  a  current  through  the  galvanom- 
eter, and  observe  the  angle  through  which  the  needle  has  turned 
when  it  comes  to  rest.  We  then  reverse  the  current  through 
the  galvanometer,  and  observe  the  corresponding  angle  of 
deflection  on  the  opposite  side  of  the  position  of  equilibrium. 
If  these  angles  are  not  equal,  we  turn  the  galvanometer  coils  in 
such  a  direction  as  to  increase  the  smaller  angle  of  deflection, 
and  repeat  until  the  difference  of  the  two  angles  is  a  small 


158 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


fraction  of  either  one  of  them.  In  doing  this  it  must  be  remem- 
bered that  if  the  scale  is  turned  with  the  galvanometer  coils,  the 
needle  will  come  to  rest  at  a  new  position  with  respect  to  the 
scale  ;  i.e.  the  galvanometer  will  have  a  new  "  zero  point." 

In  measuring  current  by  means  of  a  galvanometer,  angles  of 
deflection  should  be  determined  for  the  current,  both  direct  and 
reversed.  There  are  two  reasons  for  this  : 

(1)  If  the  galvanometer  coil  makes  a  small  angle  with  the 
magnetic  meridian,   it   may  be  proved*  that  the  mean  of  the 
deflections  for  direct  and  reversed  current  will  be  in  error  by  a 
small  quantity  of  the  second  order. 

(2)  The  equilibrium  position  or  zero  point  of  a  galvanometer 
needle   is   constantly  varying,    the   fluctuations   being   due   to 
variations  in  the  earth's  magnetic  field.     Now  it  is  quite  as 
convenient  to  observe  the  reading  for  reversed  current  as  it  is 
to  observe  the  zero  reading  for  every  measurement. 

In  measurements  by  this  method  of  direct  and  reversed 
deflections  a  commutator,  or  reversing  key,  is  used.  A  simple 

form  consists  of  a  block  of  wood 
containing  four  holes  filled  with 
mercury,  and  a  second  block  con- 
taining two  U-shaped  conductors 
of  heavy  copper  wire.  The  de- 
vice is  shown  in  Fig.  51.  The 
wires  leading  from  the  battery 
are  to  be  connected  to  mercury 
cups  at  mv  m%,  while  the  gal- 
vanometer is  connected  to  the 
remaining  cups.  If  the  second 
block  (b)  containing  the  U-shaped  conductors  be  placed  upon 
the  first  with  the  conductors  dipping  into  the  mercury  cups,  the 
current  from  the  battery  will  flow  in  one  direction  through  the 


(V 

mi 

I     i 

r    t 

ma 

}      I 

b 

b 

I.I. 

•—  • 

Fig.  51. 


*  See  Mascart  and  Joubert,  Lesons  sur  I'electricite"    et  le  magnetism,  vol.  2, 
p.  235 ;   also  Nichols,  The  Galvanometer,  Lecture  2. 


THE    ELECTRIC   CURRENT. 


159 


galvanometer,  while  the  current  through  the  galvanometer  will 
be  reversed  by  lifting  the  second  block,  turning  it  90°  about  a 
vertical  axis,  and  again  dropping  the  conductors  into  the  mer- 
cury cups. 

The  angle  of  deflection  of  the  galvanometer  needle  may 
be  determined  directly  from  the  reading  of  a  long  pointer 
moving  over  a  circular  graduated  scale.  In  this  case  both  ends 

o  „ 


i— J 


L 

\      \ 
\       \ 

l         \ 

\         \ 

\         \ 

\          \ 

\           \ 

\            \ 

\ 

\ 

s           f 

\                               X 

\                 \ 

I             A 

T 

Fig.  52. 

of  the  pointer  should  be  read  in  order  to  eliminate  eccentricity, 
as  well  as  to  get  a  more  accurate  value  of  the  deflection. 

In  many  cases  the  angle  of  deflection  is  determined  by 
means  of  a  small  mirror  permanently  attached  to  the  magnetic 
needle.  With  mirror  galvanometers  either  a  telescope  and 
scale,  or  a  lamp  and  scale,  may  be  used.  In  either  case,  the 
angle  of  deflection  is  computed  in  the  same  way. 

Let  OM  (Fig.  52)  be  the  horizontal  section  of  the  mirror 
attached  to  the  galvanometer  needle,  5  the  scale,  and  T  the 
telescope.  The  telescope  and  scale  should  be  adjusted  at 
right  angles  to  each  other,  and  so  placed  that  the  portion 


160  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

of  the  scale  immediately  below  (or  above;  the  telescope  is 
seen  reflected  from  the  mirror  through  the  telescope  when 
no  current  flows  through  the  galvanometer.  If  the  gal- 
vanometer needle  be  now  deflected  through  the  angle  B,  a 
new  portion  of  the  scale  will  be  seen  reflected  from  the 
mirror.  From  the  law  of  reflection,  the  angle  which  the  ray 
reflected  from  the  mirror  into  the  telescope  makes  with  the 
^normal  must  be  equal  to  the  angle  which  the  incident  ray 
AO  makes  with  the  same  normal.  It  follows  that  the  angle 
AOT  between  the  reflected  and  incident  rays  is  equal  to  2  S. 
If  /  is  the  deflection  along  the  scale  from  the  zero  point,  and 
L  the  distance  of  the  scale  from  the  mirror,  we  have 

tan  2  8=—-  (112) 

J-^S 

From  this  equation  and  a  table  of  tangents,  8,  and  hence  tan  S, 
may  be  deduced. 

It  may  be  readily  shown  than  when  8  is  quite  small,  tan  8  is 
very  nearly  proportional  to  /.  Under  this  condition  it  follows 
that  current  is  very  nearly  proportional  to  deflection,  and  we 
may  use  the  equation 

/  =  V>  (i  1 3). 

in  which  70  is  the  constant  per  scale  division.     The  error  in 
using  this  approximate  value  for  the  current  is  as  follows  : 

When  —  =  —  ,  the  error  is  about  0.0025. 
L     10 

When  —  =  -,  the  error  is  about  o.oioo. 
L     4 

When  —  =  -,  the  error  is  about  0.0600. 
L     2 

From  an  inspection  of  Fig.  5  2  a,  it  is  obvious  that  a  scale 
might  be  so  curved  that  deflection  of  the  ray  of  light,  as 
measured  along  the  scale,  would  be  directly  proportional  to 


THE   ELECTRIC   CURRENT. 


161 


tan  S.  This  curve  would  not  be  one  easy  to  construct,  but 
it  can  easily  be  proved  that  if  a  scale  S'  be  made  upon  the 
arc  of  a  circle  whose  radius  is  f  of  the  distance  from  the 
mirror  to  the  scale  immediately  under  the  telescope,  the  error 
in  assuming  that  tan  S  is  proportional  to  the  deflection  along 
the  scale  does  not  exceed  o.ooi  for  any  deflection  of  the 
needle  not  exceeding  24°. 

When  the  current  is  reversed  in  a  galvanometer,  it  often 
takes  several  minutes  for  the  needle  to  come  to  rest.     Time 


Fig.  52  a. 

may  be  saved  by  closing  the  circuit  as  the  galvanometer  needle 
gets  about  to  the  end  of  its  free  swing  in  the  direction  in 
which  the  current  would  deflect  it. 

There  are  two  principal  methods  of  "damping"  the  oscil- 
lations of  galvanometer  needles,  and  making  the  instrument 
nearly  or  quite  "dead-beat." 

(1)  By  the  attachment  of  a  mica  or  aluminum  vane  to  the 
needle.     This  vane,  by  friction  against  the  air  in  an  inclosed 
place,  brings  the  needle  to  rest  much  more  quickly  than  would 
otherwise   be   the   case.      The    action    may   be   increased    by 
suspending  the  vane  in  a  vessel  of  oil. 

(2)  By  suspending   the   magnetic    needle  within   a   cavity 
in  a  small  mass   of   copper.     As  the  magnet    moves   in  this 
cavity,  causing  the  lines  of  force  to  sweep  through  the  copper, 


162  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

currents  of  electricity  are  induced.  These  currents,  as  stated 
in  Lenz's  law,  are  always  in  such  a  direction  as  to  oppose 
the  motion  which  produced  them. 

The  best  example  of  this  is  found  in 
the  type  of  instrument  first  designed  by 
Siemens.  In  this  form  of  galvanometer 
the  magnetic  needle  is  of  the  horseshoe 
type,  ordinarily  called  a  bell  magnet.  This 
magnet  is  suspended  in  a  hole  but  little 
larger  than  itself  in  a  copper  sphere.  Fig. 
53  shows  a  vertical  section  of  the  cop- 
per sphere,  with  the  inclosed  magnet ; 
Fig.  54  represents  a  cross-section  of  the 
same  magnet. 

In  the  use  of  sensitive  galvanometers, 
the  question  will  often  arise  as  to  what 
type  of  galvanometer  will  be  most  sensitive 
for  a  particular  purpose.  This  question 

cannot  be  settled  in  general    terms,  on  account  of   the  great 
difference  between  the   different   types.     The  more  restricted 
question  as  to  what  number  of  turns  will  be  most  sen- 
sitive for  a  particular  purpose  may  be  easily  determined. 
If  the  type  of  galvanometer,  the  size  of  the  coils, 
the  mass  of  copper  in  them,  and  the  closeness  of  the 
turns  to  the  needle  remain  the  same,  it  may  readily 
be  proved  that  that  instrument  will  be  most  sensitive 
whose  internal  resistance  is  equal  to  the  external  resist- 
*ance  in  series  with  it. 

This  relation  may  be  demonstrated  as  follows  :  Let 
Fbe  the  volume,  L  the  total  length,  s  the  cross-section, 
Rg  the  resistance,  and  p  the  specific  resistance  of  the 
wire  to  be  used  in  the  galvanometer  coils.  Then  we 
have  Flg.  54. 

(H4) 


THE   ELECTRIC    CURRENT. 


163 


If  70  is  the  constant  per  scale  division  of  the  galvanometer, 
we  have  70  =  — ,  in  which  C  is  a  constant  depending  on  the  type 

-L>  t 

of  galvanometer. 

If  E  is  the  electromotive  force,  R  the  external  resistance, 
and  S  the  galvanometer  deflection,  we  have 


If  this  equation  is  differentiated  with  respect  to  L,  we  have 


* 

dL 


Consequently  S  will  be  a  maximum  when 

R=±L*=Rf  (116) 

a  result  that  coincides  with  the  statement  which  we  desire  to 
verify. 

EXPERIMENT  R1.     Law  of  the  tangent  galvanometer. 

If  a  galvanometer  coil  is  placed  with  the  plane  of  its  wind- 
ings in  the  magnetic  meridian,  the  magnetic  field  due  to  a 
current  circulating  in  the  coils  will  be  (in  the  axis  of  the  coil) 
at  right  angles  to  the  earth's  magnetic  field.  The  resultant  of 
these  two  fields  will  therefore  make  an  angle  with  the  magnetic 
meridian  whose  tangent  is  the  ratio  of  the  intensity  of  the 
earth's  field  to  the  field  due  to  the  current  in  the  galvanometer 
coils.  A  short  magnetic  needle  suspended  anywhere  in  the 
axis  of  the  coil  will  set  itself  along  this  resultant  direction.  If 
the  needle  is  not  a  short  one,  it  will,  when  considerably  deflected 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

from  the  meridian,  extend  beyond  the  axis  to  points  where  the 
two  components  of  the  field  are  not  at  right  angles  to  each 
other.  Therefore  the  tangent  law  will  no  longer  hold. 

This  experiment  is  intended  to  give  a  method  of  testing 
experimentally  whether  a  given  galvanometer  obeys  the  law  of 
tangents  (i.e.  whether  the  tangent  of  the  angle  of  deflection  is 
proportional  to  the  current). 

Connect  the  galvanometer  in  series  with  a  resistance  box 
and  cell,  and  place  a  reversing  key  somewhere  in  the  circuit  so 
that  the  direction  of  the  current  in  the  galvanometer  can  be 
readily  reversed.  Then  observe  the  reading  of  the  galvanom- 
eter, with  current  both  direct  and  reversed,  for  ten  or  twelve 
different  values  of  the  resistance  in  the  box,  choosing  the  resist- 
ances so  that  the  deflection  of  the  galvanometer  is  varied  from 
the  smallest  that  can  be  accurately  observed  up  to  the  largest 
that  can  be  used. 

From  the  data  thus  obtained,  plat  a  curve,  using  resistances 
as  abscissas  and  cotangents  of  deflections  as  ordinates.  If  a 
reflecting  galvanometer  is  used,  it  will  be  necessary  to  compute 
the  angle  of  deflection  from  the  linear  deflection  and  the  dis- 
tance between  mirror  and  scale.  If  the  galvanometer  obeys 
the  tangent  law,  the  curve  obtained  by  platting  as  above  should 
be  a  straight  line.  Draw  a  straight  line,  therefore,  that  passes 
as  nearly  as  possible  through  all  the  points,  and  produce  this 
line  backward  until  it  intersects  the  horizontal  axis.  The  dis- 
tance between  the  origin  and  this  point  of  intersection  is  a 
measure  of  the  resistance  in  the  circuit  outside  the  resistance 
box  ;  i.e.  if  we  call  this  resistance  R^  then  ^0  =  galvanometer 
resistance  +  battery  resistance  +  resistance  of  connecting  wires. 

Now,  if  E  is  the  E.  M.  F.  of  the  cell,  and  R  the  resistance 
in  the  box,  then 

(117) 


If  E  is  known,  and  RQ  determined  from  the  curve,  as  stated 
above,  the  constant  of  the  galvanometer  can  be  computed.     In 


THE   ELECTRIC   CURRENT. 


I65 


making  this  computation,  take  the  E.  M.  F.  of  a  gravity  cell  as 
I  volt.* 

In  some  reflecting  galvanometers  in  which  the  deflection  is 
small,  the  deflection  itself  is  nearly  proportional  to  the  current ; 
i.e.  if  the  deflection  on  the  scale  is  8,  then  7=/0S.  To  test 
whether  this  is  true  in  the  case  of  the  galvanometer  used,  plat  a 
curve  with  box  resistances  as  abscissas,  and  reciprocals  of  deflec- 
tions as  ordinates.  If  /  =  /0S,  the  line  obtained  will  be  straight. 

By  some  students,  the  following  method  of  using  the  data 
may  be  preferred.  From  the  data,  determine  the  value  of  ^0, 
as  in  Exp.  T5.  Compute  the  value  of  the  current  in  each  case 


\ 
.,[_ 

LAWC 

>F  TAN 

iENTO 

M.VANC 

METER 

^^ 

^* 

+*~ 

\ 

4I 
1 
1 

^^t^» 

l^f* 

^* 

^ 

lil 

3| 
oj 

^^- 

^^ 

^ 

8 

1 

^f^ 

^ 

^  " 

^^ 

,4- 

•"*°*'^ 

B0> 

C   RES 

STAN 

CES 

10        12        14 

Fig.  55. 


16 


IB       20       22       24       26      28 


from  Ohm's  law.     If  the  galvanometer  obeys  the  tangent*  law, 
the  ratio  of  current  to  tangent  of  deflection  will  be  a  constant. 

The  following  data  are  typical  of  the  results  to  be  obtained, 
and  will  serve  to  indicate  the  method  of  arranging  and  tabulat- 
ing readings.  The  curve  given  in  Fig.  55  shows  the  graphical 
method  of  testing  the  deviation  of  the  instrument  from  the  law 
of  tangents. 

*  The  E.  M.  F.  of  the  battery  used  in  taking  the  following  data  was  4  volts. 


1 66 


JUNIOR   COURSE   IN   GENERAL  PHYSICS. 


LAW  OF  TANGENT  GALVANOMETER. 


RESISTANCE 
IN  Box. 

GALVANOMETER  READINGS. 

MEAN 
DEFLECTION. 

tan  8 

cot  8 

/o 

Direct. 

Reversed. 

0 

r  N.  62°.5 

t    S.    62°.0 

N.  63°.o  1 
S.  63°.5  I 

62°.  70 

1-937 

0.516 

0.6  1  3 

2 

r  N.  5o°.6 
1   S.   so°.o 

N.  50°  .0  \ 
S.  so°.2  J 

50°.  20 

I.2OO 

0-833 

0.625 

4 

/  N.  4i°.5 
I    S.   4i°.o 

N.  4i°.o  | 
S.  4i°.2  / 

4I°.20 

0.875 

I.I42 

0.623 

6 

/   N.  34°.8 
\   S.   34°.2 

N.  34°.5  \ 
S.  35°.i  / 

34°-8o 

0.695 

1-439 

0.616 

10 

r  N.  250.5 

I   S.   25°.o 

N.  250.5  I 
S.  26°  .0  / 

25°-5Q 

0.477 

2.096 

0.628 

14 

f   N.  2o°.4 
1   S.    i9°.8 

N.  20°.  I  ) 

S.  20°.6  / 

20°.  20 

o.368 

2.718 

0.627 

18 

f   N.  i7°.o 
1   S.    i6°-5 

N.  i7°.o  1 
S.  i7°.6  J 

i7°.oo 

o.3o6 

3-271 

0.610 

24 

f   N.  i3°.4 
1    S.    I2°.9 

N.  130.2  ) 
S.  i3°.8  / 

I3°-30 

0.236 

4-230 

0.620 

30 

f   N.  io°.9 
1   S.    io°.3 

N.  io°.9  •) 
S.  ii°.4  J 

io°.90 

0-193 

5-193 

0.621 

50 

f   N.     7°.o 
I   S.     6°.s 

N.  70.1  1 
S.  7°.6  / 

7°-05 

0.124 

8.086 

o.6i3 

From  curve  JRQ  =  3.33  ohms. 
"         «       /0  =  0.620  amp. 

Last  column  computed  assuming  value  of  ^0  obtained  from  plat. 


EXPERIMENT  R2.     Measurement  of  current  by  electrolysis. 

One  of  the  most  accurate  methods  of  measuring  current  is 
by  means  of  the  amount  of  copper  or  silver  deposited  in  a  vol- 
tameter through  which  the  current  flows. 

The  voltameter  deposit  represents  the  integrated  value  of 
the  current  extending  over  considerable  time ;  that  is,  it  is  a 
measure  of  the  total  quantity  of  the  current  which  has  flowed 
through  the  voltameter.  This  instrument,  therefore,  can  only 
give  an  average  value  of  the  current.  On  account  of  this  and 
other  disadvantages,  the  voltameter  is  chiefly  used  to  calibrate 


THE   ELECTRIC   CURRENT. 


l67 


or  determine  the  constants  of  instruments  which   depend  for 
their  indications  on  the  magnetic  field  produced  by  the  current. 

In  this  experiment  the  spiral  coil  voltameter  devised  by 
Professor  H.  J.  Ryan  is  to  be  used.*  Two  coils  are  to  be 
prepared  for  each  cell  by  wrapping  copper  wire  on  cylindrical 
forms.  The  size  of  the  coils  depends  somewhat  on  the  strength 
of  the  current  used.  With  a  current  of  from  one  to  three 
amperes,  a  coil  made  of  one  and  a  half  meters  of  wire  of  1.5  mm. 
diameter  will  give  satisfactory  results. 

The  coils  should  be  of  about  the  same  length,  but  should 
differ  in  diameter  by  3  or  4  cm.,  so  that  the  smaller  may  be 
placed  inside  the  other  without  danger 
of  touching.  (See  Fig.  56.)  At  one 
end  of  each  coil  the  wire  is  to  be 
brought  out  parallel  with  the  axis 
for  several  inches  for  convenience  in 
making  connections.  These  two  coils 
are  to  be  used  as  the  electrodes  of  a 
voltameter  cell,  current  passing  in 
through  the  outer  coil  and  leaving 
the  cell  by  the  inner  coil.  The 
amount  of  copper  deposited  in  a 
known  time  is  then  sufficient  to 
determine  the  average  current  flow- 
ing. (One  coulomb  deposits  0.000328 
gram  of  copper.)  The  amount  of  copper  dissolved  is  always 
slightly  in  excess  of  the  amount  deposited,  and  for  various  rea- 
sons is  not  so  reliable  a  measure  of  the  current. 

In  preparing  the  gain  coils,  great  care  must  be  used  to  have 
them  thoroughly  clean.  A  wire  of  suitable  length  for  the  pur- 
pose should  be  fastened  by  one  end  and  then  cleaned  with  sand- 
paper. When  thoroughly  cleaned,  the  wire  is  coiled  upon  a 


Fig.  56. 


*  See  Ryan,  Transactions  of  the  American  Institute  of  Electrical   Engineers, 
vol.  6,  p.  322. 


i68 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


suitable  form,  the  latter  being  first  covered  with  clean  filter- 
paper.  After  cleaning  with  the  sandpaper,  the  coil  should  not 
be  touched  by  the  hand  anywhere  except  at  its  terminal.  If 
this  work  has  been  well  done,  the  coils  will  be  ready  for  use 
without  any  further  cleansing.  If  not,  pass  the  coil  through  a 
non-luminous  Bunsen  flame  to  remove  oil,  plunge  it  in  a  very 
dilute  solution  of  sulphuric  acid,  and  then  into  distilled  water. 
To  dry  the  coil  rapidly  and  without  danger  of  oxidation,  it  is 
first  rolled  on  filter  or  blotting  paper  until  only  a  thin  film  of 
water  remains.  This  is  rinsed  off  by  dipping  into  strong  alco- 
hol. After  again  rolling  on  filter-paper,  what  little  alcohol  is 
left  will  quickly  evaporate,  leaving  the  coil  dry  and  ready  for 


Fig.  57. 

weighing.  The  loss  coil  should  also  be  cleansed  with  sand- 
paper, but  it  is  unnecessary  to  use  the  precautions  that  are 
required  in  the  case  of  the  gain  coil. 

The  density  of  the  copper  sulphate  solution  should  lie 
between  i.io  and  1.18.  A  few  drops  of  sulphuric  acid  will 
improve  the  action  of  the  solution.  The  direction  of  the  current 
should  be  determined  by  a  compass  needle  before  the  voltameter 
is  placed  in  the  circuit.  The  connections  can  then  be  made 
in  such  a  way  as  to  make  the  deposit  occur  on  the  inner  coil. 

In  this  experiment  the  voltameter  is  to  be  used  in  determin- 
ing the  constant  of  a  tangent  galvanometer.  Two  voltameter 
cells  (Fig.  57)  are  used  as  a  check  on  the  weighings,  the  two 
cells  being  connected  in  series  with  the  galvanometer  and  with 
each  other.  Figure  58  gives  a  diagram  of  the  connections.  A 


THE   ELECTRIC   CURRENT. 


I69 


steady  current  is  sent  through  the  circuit  for  some  measured 
length  of  time,  and  the  strength  of  the  current  is  computed 
from  the  amount  of  copper  deposited.  The  deflection  of  the 
galvanometer  having  been  also  observed,  the  constant  is  readily 
computed. 

A  reversing  key  should  be  used  in  connection  with  the 
galvanometer,  the  construction  of  the  key  being  such  that 
the  current  through  the  galvanometer  can  be  reversed  without 
breaking  the  current  in  the  main  circuit.  Deflections,  both 
direct  and  reversed,  are  to  be  observed  at  intervals  of  two  or 
three  minutes  throughout  the  experiment. 

The  gain  coils  must  be  weighed  with  great  care,  and  placed 
in  the  solution  only  a  few  minutes  before  the  current  is  started. 
At  the  end  of  the  experiment  they 
should  be  immediately  removed,  dipped 
in  distilled  water,  and  dried,  as  described 
above.  The  second  weighing  should  be 
made  as  soon  as  possible  after  the  coils 
are  dry. 

The  constant  of    any  galvanometer 
may  be  measured  by  this  method.  Fis-  58. 

In  the  case  of  instruments,  the  sensitiveness  of  which  is  so 
great  that  currents  of  the  magnitude  adapted  to  the  voltameter 
cannot  be  measured  directly,  a  shunt  of  suitable  resistance, 
^«  (Fig.  58),  should  be  placed  across  the  galvanometer 
terminals.  The  ratio  of  the  current  in  the  voltameter  to  that 
which  flows  through  the  coils  of  the  galvanometer  can  readily 
be  computed. 

The  following  table  gives  the  results  obtained  in  the  cali- 
bration of  a  tangent  galvanometer,  and  shows  the  method  of 
arranging  them  : 


170  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

GALVANOMETER  CONSTANT  BY  COPPER  VOLTAMETER. 


GALVANOMETER 
READINGS. 

Other  Data  and  Results 

TIME. 

Current 
Direct. 

Current 
Reversed. 

Distance  of  mirror  from  scale  —  50,       scale 
Average  double  deflection  =  32.27,          ' 

div. 

tan  28  —  o  ^227                   25  —  17°  C'*' 

hr.  mm. 

5  =  8°  56^'              tan  5  =  0.1573 

9  49 

Circuit 

completed. 

51 

— 

42.40 

Two  voltameter  cells  in  series  : 

54 

74.62 

— 

Before.                After. 

Gain. 

57 

10      I 

5 

74.62 

42.43 
42-43 

Cathode  A  ...     27.434            28.686 
"        B  .     .     .     27.5715           28.624 

1.2520 
1.2525 

9 

74.68 

— 

Duration  of  run  =  3600  sec. 

13 

— 

42.39 

17 

74.70 

Intensity  of  current  /  =  1.059  amp. 

21 

— 

42.38 

For  tangent  galvanometer,  /=  70tan5; 

25 

74.67 

— 

/.  70  =  6.73  amp. 

29 

— 

42.32 

33 

74.70 

— 

Galvanometer,  one  turn,  needle  at  center  : 

37 

— 

42.31 

Diameter  of  ring  =77.7  cm. 

45 

74.57 

42.30 

True  constant  G  —  0.1617. 

49 

74.56 

— 

/-I0^;     .;ff=  0.109. 

49 

Circuit 

broken. 

G 

* 

EXPERIMENT  R3.  Measurement  of  the  constant  of  a  sensitive 
galvanometer. 

It  is  frequently  impracticable  to  calculate  the  constant  of  c. 
sensitive  galvanometer  from  its  dimensions  and  from  the  value 
of  the  horizontal  intensity  of  magnetism  at  the  point  where  the 
needle  hangs.  The  constant  of  such  a  galvanometer  can  best 
be  determined  by  measuring  the  deflection  of  the  needle  which 
a  known  current  produces.  The  constant  can  then  be  deter- 
mined from  one  of  the  equations  : 

/=/0tanS, 

7=/0sinS,  (nS) 

7=/0S, 
according  to  the  law  of  the  galvanometer. 


THE   ELECTRIC   CURRENT.  ljl 

There  are  three  principal  methods  of  determining  the  con- 
stant of  such  a  galvanometer,  depending  upon  the  method  of 
determining  the  current  flowing  through  the  galvanometer  coils. 

I. 

The  current  may  be  measured  by  means  of  a  tangent 
galvanometer  whose  constant  is  already  known.  For  this  pur- 
pose it  will  be  necessary  to  put  a  shunt  across  the  terminals  of 
the  sensitive  galvanometer,  since  the  latter  will  usually  be 
very  much  more  sensitive  than  the  instrument  whose  constant 
is  already  known. 

The  method  of  procedure  is  as  follows  : 

(1)  Connect   the   tangent   galvanometer   in    series   with    a 
battery  of  constant  E.  M.  F.,  a  reversing  key,  and  a  variable 
resistance. 

(2)  Connect  the  sensitive  galvanometer  so  that  it  shall  be  in 
multiple  with  a  portion  of  the  variable  resistance,  as  in  Fig.  59. 
The  variable   resistance   should 

be  so  adjusted  that  the  deflection 
of  the  tangent  galvanometer  is 
the  most  suitable  (about  45°,  if 
the  deflection  of  the  needle  is 
read  directly,  or  nearly  as  large 
a  deflection  as  the  scale  will  per- 
mit, if  the  reading  is  made  by  Fig.  59. 
means  of  a  mirror). 

(3)  Adjust   the   resistance   in    multiple  with   the  sensitive 
galvanometer,  until   its   deflection  is  nearly  across   the   scale. 
Observe  the  readings  of  both   galvanometers   for   direct   and 
reverse  current,  and  repeat  these  observations   several   times 
to   get    a  good   average.      As   a    check,    take    another    series 
of  observations  with  a  different  resistance  in  multiple  with  the 
sensitive   galvanometer,    producing   a   deflection   varying    con- 
siderably from  the  first. 

From  the  deflection  of  the  tangent  galvanometer,  the  cur- 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


rent  flowing  in  the  main  circuit  is  known ;  and  from  the  law  of 
divided  circuits,  the  fraction  of  the  current  flowing  through  the 
sensitive  galvanometer  can  be  computed,  provided  the  ratio  of 
the  galvanometer  resistance  to  the  shunt  resistance  is  known. 


II. 

If  the  tangent  galvanometer  be  replaced  by  a  voltameter, 
the  current  in  the  main  circuit  can  be  measured  as  in  Exp. 
R<L,  The  rest  of  the  experiment  is  the  same  as  above. 

It  may  happen  with  a  very  sensitive  galvanometer  that  a 
current  strong  enough  to  produce  a  suitable  deposit  in  the  vol- 


Fig.  60- 


tameter  will  require  a  shunt  of  excessively  low  resistance  in  mul- 
tiple with  the  galvanometer.  Under  these  circumstances,  it 
will  be  advisable  to  connect  the  galvanometer  in  multiple  with 
a  branch  which  itself  is  in  multiple  with  a  portion  of  the  main 
circuit,  as  in  Fig.  60. 

If  the  ratio  of  the  resistance  in  each  pair  of  branches  is 
known,  the  current  flowing  in  the  coils  of  the  galvanometer 
may  be  calculated  from  the  current  flowing  in  the  main  circuit. 

It  is  to  be  observed  that  both  of  the  above  methods  deter- 
mine the  constant  independently  of  the  value  of  any  resistance. 
They  simply  depend  upon  a  knowledge  of  the  ratio  of  resist- 
ances. 


THE   ELECTRIC   CURRENT. 


III. 


173 


The  current  flowing  may  be  determined  from  Ohm's 
law,  provided  the  E.  M.  F.  of  the  battery  is  known  in  volts, 
and  the  resistance  of  each  portion  of  the  circuit  is  known 
in  ohms. 

For  the  purpose  of  this  determination,  a  standard  Daniell 
cell  may  be  very  easily  constructed  as  follows :  Place  an 
amalgamated  zinc  rod  in  a  porous  cell  containing  a  saturated 
solution  of  zinc  sulphate.  Coil  around  the  porous  cell  eight 
or  ten  turns  of  rather  large  copper  wire,  which  has  been 
previously  cleaned  with  sandpaper.  Place  the  porous  cell  in 
a  larger  vessel  containing  a  semi-saturated  solution  of  copper 
sulphate.  The  two  vessels  should  be  thoroughly  cleaned 
before  using. 

Such  a  cell  at  15°  has  an  E.  M.  F.  of  1.074  volts.  It  should 
be  used  immediately,  although  its  E.  M.  F.  will  change  very 
little  for  several  hours.  The  internal  resistance  of  such  a 
cell  is  usually  negligible  compared  with  10,000  ohms.  But 
if  it  is  thought  desirable  to  do  so,  its  resistance  may  be  after- 
wards determined  by  the  method  described  in  Exp.  T6.  The 
Clark  cell  affords  a  more  accurate  standard,  but  it  is  more 
difficult  to  construct,  and  it  possesses  the  disadvantage  of 
a  high  internal  resistance. 

The  following  is  the  procedure  : 

(1)  Connect  the  galvanometer  in  series  with  a  resistance 
of  at  least  10,000  ohms,  and  a  Daniell  cell,  or  a  Clark  cell. 

(2)  Observe  the  galvanometer  readings,  and  repeat  them 
several  times  to  get  a  good  average.     As  a  check,  repeat  these 
observations  with  two  or  three  different  resistances  in  series 
with  the  cell  and  galvanometer.     The  galvanometer  deflection 
may  be  deduced  from   the  readings,  and  the  current  flowing 
may  be  calculated  from  Ohm's  law.     An  application  of  one  of 
the  above  equations  will  then  give  the  galvanometer  constant. 

It  may  happen  that  the  galvanometer  used  is  so  sensitive 


174  JUNIOR   COURSE    IN   GENERAL   PHYSICS. 

that  the  deflection  is  too  great  to  be  read  even  when  all  the 
available  resistance  is  in  the  circuit.  In  this  case  the  deflection 
may  be  diminished  as  in  I  above.  In  this  case  it  will  be 
better  to  make  the  observations  for  the  check  by  varying  the 
resistance  in  multiple  with  the  galvanometer.  This  may  be 
done  simply  by  shifting  the  points  at  which  the  galvanometer 
is  connected  to  the  main  circuit. 

In  the  above  it  has  been  assumed  that  the  law  of  the 
galvanometer  is  known.  In  nearly  all  galvanometers,  some 
one  of  the  equations  given  above  hold  pretty  accurately  up 
to  10°  or  20°,  which  is  the  maximum  deflection  that  should 
be  used  with  reflecting  galvanometers.  If  the  galvanometer 
deflection  is  read  directly  by  means  of  a  pointer  moving  over 
a  graduated  scale,  the  maximum  deflection  may  be  much 
greater.  In  this  case  current  may  not  be  at  all  proportional 
to  the  tangent  of  deflection. 

In  all  such  cases  the  galvanometer  should  be  calibrated. 
For  this  purpose  proceed  as  in  any  of  the  above  experiments, 
and  observe  the  resistances  that  correspond  to  deflections, 
varying  by  approximately  equal  increments  from  zero  to  the 
maximum  reading  that  the  scale  admits.  At  least  ten  or 
twelve  such  observations  should  be  taken. 

Plat  a  curve  with  currents  flowing  through  the  galvanom- 
eter as  abscissas,  and  galvanometer  deflections  as  ordinates. 
This  curve  is  called  the  calibration  curve  of  the  instrument. 

If  the  galvanometer  is  furnished  with  a  regulating  magnet, 
its  exact  position  should  be  noted  at  the  time  of  performing 
the  experiment. 

Addenda  to  the  report: 

(1)  Indicate  the   difficulties  which   make   it   impracticable 
to   calculate   the   constant   of   a   sensitive   galvanometer  from 
its  dimensions. 

(2)  Calculate  the   constant  of   the   instrument    considered 
as  a  tangent  galvanometer.     From  this  constant  and  the  hori- 


THE   ELECTRIC   CURRENT.  175 

zontal   intensity   of   the   field    where   the    magnet   hangs,    cal- 
culate the  true  constant  of  the  galvanometer. 

(3)  If   the  instrument  is  a  reflecting   galvanometer,  calcu- 
late the  constant  per  scale  division. 

(4)  Discuss  the   influence  of   the   position  of  a  regulating 
magnet  upon  the  sensitiveness  of  the  galvanometer.     Where 
should  the  magnet  be  placed  in  order  to  change  the  zero  point 
by  a  few  scale  divisions  and  yet  have  the  least  effect  in  chang- 
ing the  galvanometer  constant  ? 

(5)  How  could  a  magnet  be  placed  quite  near  the  galvanom- 
eter and  yet  have  an  inappreciable  effect,  either  to  turn  the 
needle,  or  to  change  the  galvanometer  constant  ? 

EXPERIMENT  R4.     Theory  of  shunts. 

Whenever  a  current  flows  in  a  divided  circuit  in  which 
there  is  no  E.  M.  F.,  the  currents  in  the  branches  are  inversely 
as  the  resistances  in  those  branches.  This  relation  may  be 

stated  as  follows  : 

Is-.I9  =  Rg:Rs.  (119) 

The   object   of  this  experiment   is   to  verify  this   relation. 
The  procedure  is  as  follows  : 

(1)  Connect  a  resistance  box  in  series  with  a  gravity  battery 
of  one  or  more  cells. 

(2)  Connect  the  terminals  of   the 
resistance  box  through  a  reversing  key 
to  a  galvanometer.     (See  Fig.  61.) 

(3)  Insert   in  the    main   circuit   a 
high  resistance  (100  or  more  times  as 
great   as   Rg).      Under   these    circum- 
stances, the  current  in  the  main  circuit 

may  be   assumed    to  be  constant,   no  -*VvV 

matter  how  much  the  resistance  in  the 

resistance  box  is  varied.     If  /  stands  for  current  in  the  main 
circuit,  the  above  equation  to  be  verified  becomes 

IgRs  +  IgRg  =  IRs,  (120) 

in  which  the  variables  are  Rs  and  Ig. 


JUNIOR   COURSE    IN   GENERAL   PHYSICS. 

(4)  Observe  the  reading  of  the  galvanometer  with  current, 
both  direct  and  reversed,  for  a  number  of  different  resistances 
in  the  box,  including  the  reading  when  the  circuit  is  broken 
through  the  box.  This  last  reading  obviously  represents  the 
constant  current  in  the  main  circuit.  The  resistances  taken 
should  be  such  as  to  make  the  galvanometer  readings  vary  by 
approximately  equal  steps. 

If  the  resistance  of  the  galvanometer  (including  connecting 
wires  in  multiple  with  the  resistance  box)  be  now  measured, 
sufficient  data  will  be  obtained  to  make  a  number  of  verifica- 
tions of  the  above  equation.  As  current  appears  in  every  term 


Fig-  62- 


in  the  first  degree,  any  quantity  which  is  proportional  to  current 
may  be  substituted  for  it ;  as,  galvanometer  deflection,  or  its 
tangent. 

If  the  observations  be  platted  with  box  resistances  as 
abscissas,  and  tangents  of  galvanometer  deflections  as  ordinates, 
the  curve  obtained  will  be  a  hyperbola,  with  an  asymptote 
parallel  to  the  axis  of  abscissas.  (See  Fig.  62.)  Knowing  the 
resistance  of  the  galvanometer,  the  constant  current  in  the 
main  circuit  may  be  calculated  from  the  co-ordinates  of  any 
point  on  the  curve.  If  the  values  thus  obtained  are  equal,  the 
above  equation  is  verified. 

If  the  observations  be  platted  with  'reciprocals  of  box  re- 
sistances as  abscissas,  and  cotangents  of  deflections  as  ordi- 


THE   ELECTRIC   CURRENT. 


177 


nates,  the  resulting  curve  should  be  a  straight  line,  the  intercept 
on  the  axis  of  abscissas  being  equal  to  —  — — 

R9 

The  following  table  gives  a  typical  set  of  data  from  such 
readings,  and  shows  the  method  of  arranging  them.  If  these 
results  be  platted  as  indicated  above,  they  will  be  found  to  give 
a  curve  the  form  of  which  is  that  of  Fig.  -62. 

TABLE. 


Galvanometer  Readings. 

Galvanometer 

Resistance 

Deflection 

in 
cv..int. 

Current 

Current 

Proportional 

Other  Data  and  Results. 

onunt. 

Direct. 

Reversed. 

to  Current. 

00 

66.00 

10.95 

55-°5 

When  circuit  broken  through 

20 

64.60 

12.25 

52.35 

shunt,                        Ig=  /=5  5.  05 

5 

61.20 

15.64 

45.56 

From  curve                          Rg  =  1.04 

3 

2 

58.50 

56.08 

18.00 
19.85 

40.50 
36-23 

Values  of  /  computed  from  points  on 

I 

51.40 

2443 

26.97 

curve. 

0.6 

48.10 

27-95 

20.15 

7,  =  8              7=54-2 

o-3 

44.15 

31.80 

12.35 

7,  =  16            7=55.6 

0.2 

42.40 

33-6o 

8.80 

7^  =  30            7=55.0 

O.I 

40.26 

35-55 

4.71 

7,7  =  40            7=54.7 

Addenda  to  the  report : 

(1)  Calculate  the  current  in  the  main  circuit  from  several 
points  on  the  curve  which  are  not  observed  points. 

(2)  From  the  curve  platted,  find  what  must  be  the  resist- 
ance of  a  galvanometer  shunt  so  that  the  current  in  the  galvan- 
ometer will  be  J,  \,  -^  of  the  total  current  in  main  circuit. 

EXPERIMENT  R5.     Applications  of  the  galvanometer  to  the 

measurement  of  current. 

I. 

Measurement  of  the  current  from  a  battery  with  different 
arrangements  of  the  cells. 

For  this  experiment  a  tangent  galvanometer  of  small 
sensitiveness  and  of  very  low  resistance  is  required.  The 


VOL.  I  —  N 


178  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

galvanometer  should  have  two  or  three  separate  coils,  giving 
different  degrees  of  sensitiveness. 

(1)  Connect  a  closed-circuit  battery*  of  four  or  six  cells 
in  series  with  the  tangent  galvanometer.     The  circuit  should 
contain  also  a  variable  known  resistance  and  a  reversing  key. 

(2)  Measure  the  current  for  several  different   resistances, 
ranging   from   zero   to   some    resistance   that  will   reduce  the 
current  to  a  quarter  or  less  of  its  original  value. 

(3)  Make  a  similar  series  of  observations  for  three  or  four 
different    groupings    of    the    cells.       For   each    current   to    be 
measured,    that   galvanometer   coil   should   be   used   that   will 
give  the  greatest  deflection,   provided  that  the   latter  is   not 
much  greater  than  50°. 

(4)  Measure  the  resistance  of  the  galvanometer  coils  and 
of  the  connecting  wires.     Plat  curves  with  resistances  outside 
of  the  battery  as  abscissas,  and  currents  in  amperes  as  ordinates. 
From  the  curves  determine  under  what  conditions  each  separate 
grouping  of  the  cells  would  produce  a  greater  current  than 
any  other  grouping. 

II. 
Measurement  of  current  by  the  Vienna  method. 

This  method  is  usually  employed  with  currents  so  large 
that  they  cannot  be  measured  directly  by  ordinary  galvanom- 
eters or  ammeters.  The  main  current  is  sent  through  a  heavy 
wire  of  German  silver  or  similar  material,  whose  resistance 
changes  very  little  with  temperature,  and  a  galvanometer  is 
connected  in  multiple  with  this  resistance.  A  constant  small 
proportion  of  the  current  will  always  pass  through  the  galva- 
nometer, and  can  be  measured.  From  the  resistance  of  the 
German  silver  coil,  together  with  that  of  the  galvanometer, 
the  ratio  of  this  measured  current  to  the  total  current  can 
be  computed,  and  the  latter  is  therefore  determined. 

*  Any  battery  which  does  not  suffer  marked  polarization  will  serve  for  this 
purpose. 


THE   ELECTRIC   CURRENT. 

The  practice  of  the  method  may  be  illustrated  by  the 
measurement  of  the  current  from  a  non-polarizing  cell  of  high 
electromotive  force  and  low  internal  resistance.  In  place  of 
a  cell  a  commercial  thermo-battery  may  be  used. 

Before  beginning  the  experiment,  compute  the  resistance 
of  the  shunt  that  must  be  put  across  the  terminals  of  the 
galvanometer,  in  order  that  the  maximum  readable  galvanom- 
eter deflection  will  be  produced  when  the  maximum  current 
to  be  measured  flows  in  the  main  circuit.  This  can  be  done 
if  the  galvanometer  constant  is  known  ;  but  it  will  be  necessary 
to  assume  approximate  values  for  the  electromotive  force  and 
internal  resistance  of  the  battery. 

Having  determined  the  proper  resistance  of  the  shunt, 
proceed  as  follows  : 

(1)  Connect  the  cell  in  series  with  a  variable  resistance, 
and  with  the  shunt  which  is  in  multiple  with  the  galvanometer. 

(2)  Observe  the  galvanometer  readings  when  several  dif- 
ferent  resistances   are   used   in   series  with  the   cell.      These 
resistances  should  vary  from  one  to  ten  ohms. 

(3)  Plat  a  curve  with  resistances  as  abscissas,  and  reciprocals 
of   currents   flowing   in   the   main   circuit  as  ordinates.     This 
curve    should   be  a  straight  line,  and  from  its  constants  the 
electromotive   force   and   internal   resistance   of   the  cell   may 
be  computed. 

III. 

To  investigate  the  effect  of  polarization  upon  current, 

(1)  Take  a  Le  Clanche  cell,  or  some  other  cell  that  polarizes 
rapidly,  and  connect  it  in  series  with  five  or  ten  ohms'  resistance. 

(2)  Connect  a   sensitive   galvanometer  whose   constant   is 
known  in  multiple  with  a  portion  of  this  resistance,  such  that 
the  galvanometer  deflection  is  quite  large. 

(3)  Observe   the    galvanometer   readings    both    direct   and 
reversed  every  three  or  four  minutes  for  half  an  hour  or  longer. 
Then  break  the  circuit,  stir  the  solution  in  the  cell,  and  in  the 


180  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

course  of   ten   or  fifteen   minutes   close   the   circuit,   measure 
the  current  flowing,  and  repeat  three  or  four  times. 

(4)  Take  another  cell  as  nearly  as  possible  like  the  first 
one,  and  make  a  similar   series   of   observations,  but  with   a 
resistance  of  50  or  100  ohms  in  series  with  it. 

(5)  Compute  the  current  flowing  in  the  main  circuit,  and 
plat  a  curve  for  each  cell,  with  times  as  abscissas  and  currents 
as  ordinates. 


CHAPTER   VII. 

GROUP  S:    DIFFERENCE  OF  POTENTIAL  AND  ELECTRO- 
MOTIVE FORCE. 

(S)  General  statements ;  (Sx)  Comparison  of  two  electromotive 
forces ;  (S2)  Ohms  method  for  the  measurement  of  the 
E.  M.  F.  of  a  battery  ;  (S3)  Potential  difference  at  the  termi- 
nals of  a  battery  as  a  function  of  the  external  resistance ; 
(S4)  Fall  of  potential  in  a  wire-carrying  current ;  (S6)  Beetz 
method  of  measuring  electromotive  forces ;  (S6)  Lines  of 
equal  potential  in  a  liquid  conductor ;  (S7)  Variation  in  the 
E.  M.  F.  of  a  thermo-element  with  change  of  temperature. 

(S).  General  statements  concerning  difference  of  potential  and 
electromotive  force. 

The  indiscriminate  use  of  the  terms  "electromotive  force"  and 
"difference  of  potential"  has  given  rise  to  much  confusion.  The 
following  treatment  of  the  subject,  though  different  from  that 
of  many  writers,  is  believed  to  be  entirely  consistent  with  the 
facts.  Moreover,  it  is  hoped  that  it  will  make  clear  to  the  mind 
of  the  student  the  relation  between  two  ideas  which,  though 
intimately  related,  are  nevertheless  entirely  distinct. 

The  difference  of  potential  between  two  points  is  that  differ- 
ence in  condition  which  tends  to  produce  a  transfer  of  electrifi- 
cation from  one  point  to  the  other  point.  The  measure  of  this 
difference  of  potential  is  the  amount  of  work  that  would  be  done 
by  or  against  electrical  forces  in  carrying  unit  quantity  of  elec- 
tricity from  the  one  point  to  the  other  point. 

Any  generator  of  electricity  (whether  it  be  a  battery,  dynamo, 
or  electrical  machine)  is  capable,  when  energy  is  supplied  to  it, 

181 


182  JUNIOR   COURSE   IN   GENERAL    PHYSICS. 

of  maintaining  a  difference  of  potential  between  its  terminals, 
even  though  they  are  connected  by  a  conductor.  It  is  to  this 
capability  of  maintaining  a  difference  of  potential,  that  we  apply 
the  name  of  electromotive  force.  The  electromotive  force  of  a 
generator  is  measured  by  the  maximum  difference  of  potential 
which  it  is  capable  of  producing  when  no  current  flows.  Or, 
when  a  current  is  allowed  to  flow,  it  is  measured  by  the  differ- 
ence of  potential  at  the  terminals,  plus  the  fall  of  potential  due  to 
the  resistance  of  the  generator. 

From  these  definitions  it  follows  : 

(1)  That   there  is  a   difference   of   potential  between   any 
two  points  of  a  circuit  conveying  a  current. 

(2)  That   the   electromotive   force   of    a   circuit    is   always 
located  in  the  generator.      The    source  of   a  counter   electro- 
motive   force    may   always    be    looked    upon    as    a    negative 
generator.     So  far  as  our  present  knowledge  extends,  there  is 
never   any   electromotive    force    in    a   perfectly   homogeneous 
conductor   which    is    not    moving    relatively    to    a    magnetic 
field. 

The  above  meaning  of  the  term  "electromotive  force  "  is  always 
in  mind  when  it  is  said  that  a  given  conductor  is  the  seat*  of  an 
electromotive  force,  as  in  the  case  of  a  wire  moving  in  a  mag- 
netic field  ;  also  when  it  is  said  that  there  is  no  electromotive 
force  in  a  given  branch  of  a  multiple  circuit.  Counter  electro- 
motive force  is  the  true  negative  of  electromotive  force  as  above 
defined. 

Ohm's  law  as  originally  stated,  using  modern  terms,  is :  The 
current  flowing  in  a  (perfectly  homogeneous)  conductor  (not  moving 
relatively  to  a  magnetic  field}  is  directly  proportional  to  the  dif- 
ference of  potential  between  the  terminals  of  the  conductor.  If  the 
conductor  between  two  points  is  in  any  way  varied  subject  to 
the  above  conditions,  the  current  will  be  equal  to  the  difference 
of  potential  between  the  points  divided  by  a  quantity  known  as 

*  See  Gray's  Absolute  Measurements  in  Electricity  and  Magnetism,  pp.  142-146. 


POTENTIAL  AND  ELECTROMOTIVE  FORCE. 

the  resistance  of  the  conductor  between    the   points, 
which  we  have 


or      = 


dR 


From 


(121) 


The  statement  that  the  current  flowing  in  a  circuit  is  equal 
to  the  total  electromotive  force  in  the  circuit,  divided  by  the 
total  resistance  of  the  circuit,  is  a  deduction  from  Ohm's  law.* 


270 


The  above  discussion  may  be  fixed  in  the  mind  of  the 
student  by  the  following  graphic  treatment  of  two  particular 
cases. 

Let  BC,  Fig.  63  a,  be  the  two  poles 
of  a  cell  whose  electromotive  force  is 
two  volts,  and  internal  resistance  four 
ohms.  The  negative  pole  of  the  cell  is 
maintained  at  zero  potential,  by  being 
grounded,  and  the  two  poles  are  connected  by  a  homogeneous 
conductor  of  twelve  ohms'  resistance :  CA  four  ohms,  and  AB 
eight  ohms. 

If  a  curve  be  platted  showing  the  relation  between  potential 
and  resistance,  with  resistances  counting  from  A  in  the  di- 


Fig.  63  a. 


*  See  Gray's  Absolute  Measurements  in  Electricity  and  Magnetism,  pp.  142-146. 


1 84  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

rection  of  the  current  as  abscissas  and  potentials  for  ordi- 
nates,  the  result  will  be  as  given  in  Fig.  63.  From  A  to  B 
the  potential  falls  uniformly;  between  the  negative  pole  and 
the  liquid  there  is  a  finite  difference  of  potential  represented 
by  BB' ;  in  the  liquid,  supposed  homogeneous,  there  is  a  fall 
of  potential,  at  the  same  rate  as  in  the  outside  conductor ; 
between  the  liquid  and  the  positive  pole  there  is  a  finite 
difference  of  potential  represented  by  CfC,  and  from  C  to  A 
the  potential  falls  at  the  same  rate  as  before,  reaching,  of 
course,  the  original  value. 

By  the  application  of  Ohm's  law  the  current,  1=-=,  may 

dR 

be  derived  from  any  part  of  the  circuit  that  is  homogeneous. 
The  result  is  obviously  the  same  whether  increments  of 
potential  and  resistance  are  infinitesimal  or  of  any  magnitude. 
If  the  conductor  is  cut  at  At  Fig.  63  a,  then  the  potential  of 
AB  will  immediately  fall  to  zero.  As  no  current  flows,  there  will 
be  no  fall  of  potential  in  the  liquid  ;  the  potential  of  C  will  there- 
fore immediately  rise  by  the  amount  of  the  former  fall  through 
the  liquid.  The  broken  line  represents  the  potential  in  the 
cell  and  conductor  after  the  circuit  is  broken.  The  electro- 
motive force  of  the  cell  E  is  measured  by  the  maximum 
difference  of  potential  between  its  terminals  when  no  current 
flows.  This  is  obviously  equal  to  pd+pd{ ',  in  which  pd  is  the 
difference  of  potential  between  the  terminals  before  the  circuit 
is  broken,  and  pd'  is  the  fall  of  potential  in  the  cell  due  to 
its  resistance.  It  is  obvious  from  the  geometry  of  the  figure 

that  — ,  in  which  R  is  the  total  resistance  of  the  circuit,  gives 

dV 

the  same  value  for  the  current  as  — -  taken  in  any  homogeneous 

dR 

part  of  the  circuit. 

Figure  64  represents  the  potential  as  a  function  of  the 
resistance  in  a  circuit,  in  which  the  generator  is  a  dynamo. 
In  this  case  the  potential  is  not  a  discontinuous  function  of 
the  resistance.  The  electromotive  force  is  not  located  at  a 


POTENTIAL   AND   ELECTROMOTIVE   FORCE.  185 

point  (or  at  a  surface)  in  the  circuit  as  in  the  case  of  a  cell, 
but  it  exists  in  all  those  parts  of  the  armature  which  cut 
lines  of  force.  With  this  exception,  the  above  discussion 
applies  to  the  present  case,  word  for  word.  If  the  circuit 
is  broken  as  before,  the  broken  line  shows  the  condition 
of  affairs,  provided  that  the  resultant*  magnetic  field  remains 


2.0 


1.5- 


2,0- 


.5- 


dV 


fid 


pd 


12 


16 


8 

OHMS 

Fig.  64. 

unchanged  after  the  circuit  is  broken.  This  assumption  does 
not  hold  true,  of  course,  in  the  case  of  the  series-wound 
dynamo. 

The  above  graphic  representations  of  the  potentials  in  a 
conductor  carrying  a  current,  bring  prominently  forward  the 
fact  that  in  a  conductor  not  containing  an  electromotive  force 
the  current  always  flows  from  points  of  higher  potential  to 
points  of  lower  potential ;  but  that  in  a  conductor  or  in  that 
part  of  it  containing  an  electromotive  force  producing  a  current, 
the  current  always  flows  from  points  of  lower  to  points  of  higher 
potential. 


*  By  resultant  field  is  meant  the  field  that  is  the  resultant  of  the  field  due  to  the 
field  magnets  and  to  the  current  in  the  armature  coils. 


1  86  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

The  figure  also  illustrates  the  fact  that  the  electromotive 
force  in  a  circuit  is  a  constant  which  is  independent  of  the 
resistance  of  the  generator  and  of  the  external  circuit,  while  the 
difference  of  potential  between  two  points  is  not  independent  of 
those  quantities. 

In  the  case  given,  the  electromotive  force  is  two  volts,  but  the 
greatest  difference  of  potential  between  any  two  points  is  1.5 
volts.  If  the  cell  be  replaced  by  two  cells  of  half  the  electro- 
motive force  and  half  the  internal  resistance  separated  from 
each  other  by  six  ohms'  resistance,  the  electromotive  force  in 
the  circuit  would  be  two  volts,  but  the  greatest  difference  of 
potential  between  any  two  points  would  be  only  0.75  volt. 
So  we  might  imagine  a  circuit  in  every  part  of  which  there 
was  an  electromotive  force  directly  proportional  to  the  resist- 
ance of  that  part.  The  electromotive  force  and  current  would 
still  be  the  same,  but  there  would  be  no  difference  of  potential 
between  any  two  points  of  the  circuit.  In  any  part  of  such  a 
circuit,  the  difference  of  potential  due  to  its  resistance  is  exactly 
balanced  by  the  electromotive  force  generated  in  that  part. 
This  does  not  invalidate  Ohm's  law  as  originally  stated,  for 
that  law  is  no  longer  applicable. 

By  way  of  illustration,  imagine  a  perfectly  uniform  and 
homogeneous  ring  moving  with  respect  to  a  uniformly  magne- 
tized cylindrical  bar  magnet,  their  axes  remaining  coincident. 
This  would  constitute  a  circuit  containing  an  electromotive  force 
and  carrying  a  current,  but  in  which  there  would  be  no  differ- 
ence of  potential. 

From  definitions  it  follows  that  whenever  electricity  is 
transferred  along  a  circuit,  work  is  done  between  the  points 
a  and  b,  according  to  the  relation 


The  difference  of  potential  between  a  and  b  is  one  electro- 
magnetic unit,  if  work  is  done  at  the  rate  of  i  erg  per  second, 
when  the  current  flowing  is  one  electromagnetic  unit.  This 


POTENTIAL  AND  ELECTROMOTIVE  FORCE.      ^7 

choice  of  unit  potential  difference  makes  k  unity  in  the  above 
equation.  The  electromagnetic  unit  of  electromotive  force  is 
that  electromotive  force  which  is  capable  of  producing  unit 
difference  of  potential.  It  may  be  proved  that  the  electromag- 
netic unit  of  electromotive  force  is  produced  whenever  unit 
magnetic  lines  of  force  are  cut  at  the  rate  of  one  per  second. 
The  practical  unit  of  difference  of  potential  is  called  a  "volt.'* 
It  is  equal  to  io8  electromagnetic  units. 

EXPERIMENT  Sx.     Comparison  of  two  electromotive  forces. 

I. 

From  Ohm's  law  we  have,  if  a  tangent  galvanometer  is  used, 

.  (I22) 


If  the  two  cells  whose  E.  M.  F.'s*  are  to  be  compared  are 
allowed  to  send  a  current  successively  through  the  same 
resistance,  the  ratio  of  their  E.  M.  F.'s  will  be  equal  to  the 
ratio  of  the  tangents  of  the  angles  of  deflection.  To  perform 
the  experiment  we  proceed  as  follows  : 

(1)  Connect  one  of  the  cells  in  series  with  a  sensitive  gal- 
vanometer and  a  high  resistance  (500  or  more  times  as  great  as 
the  resistance  of  the  cell). 

(2)  Observe   the   galvanometer   readings    (both   direct   and 
reversed). 

(3)  Repeat  these  observations  with  the  other  cell  connected 
in  series  with  the  same  resistance.     A  number  of  independent 
determinations  of  the  ratio  of  the  two  E.  M.  F.'s  may  be  made 
by  varying  the  high  resistance  in  the  circuit  ;    or  by  varying 
the  sensitiveness  of  the  galvanometer  by  means  of  a  controlling 
magnet. 


*In  this  and  subsequent  directions  for  the  performance  of  experiments,  this 
well-known  abbreviation  will  be  used  for  electromotive  force. 


1 88  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

II. 

The  method  above  described  assumes  that  the  resistance  of 
each  cell  is  negligible  compared  with  the  total  resistance  of  the 
circuit.  If  this  is  not  the  case,  the  following  method  is  always 
applicable. 

When  the  two  cells  are  connected  so  as  to  assist  each 
other,  the  two  electromotive  forces  are  added ;  when  they  are 
connected  so  as  to  oppose  each  other,  the  current  in  the  circuit 
is  that  due  to  the  difference  of  the  two  E.  M.  F.'s.  Since 
the  resistance  of  the  circuit  is  the  same  in  both  cases, 
the  two  currents  observed  are  proportional  to  the  sum  and 
difference  respectively  of  the  electromotive  forces.  The  two 
currents  being  deduced  from  the  galvanometer  deflection, 
the  ratio  of  the  two  electromotive  forces  can  be  computed. 

If  it  is  not  known  which  cell  has  the  greater  electromotive 
force,  it  should  be  observed  whether  the  deflection  is  reversed 
or  simply  lessened  when  the  terminals  of  the  given  cell  are 
reversed. 

The  cells  are  first  connected  in  series  with  a  galvanom- 
eter, and  such  a  resistance  is  placed  in  the  circuit  as  will 
make  the  deflection  one  that  can  easily  be  read.  We  then 
reverse  one  of  the  cells  so  that  the  two  act  against  one  another, 
everything  else  about  the  circuit  remaining  the  same,  and 
observe  the  deflection. 

A  number  of  independent  determinations  should  be  made  as 
described  above. 

If  the  E.  M.  F.'s  are  very  nearly  equal,  great  care  should 
be  exercised  in  reading  the  galvanometer  when  the  cells  are 
opposed  to  each  other.  It  is  better,  in  such  a  case,  to  use 
two  cells  of  the  kind  having  the  smaller  E.  M.  F.  against 
one  of  the  other  kind. 

If  this  method  be  used  with  cells  either  of  which  polarizes 
rapidly,  it  must  not  be  expected  that  the  results  of  a  series  of 
observations  will  be  entirely  concordant. 


POTENTIAL   AND   ELECTROMOTIVE   FORCE. 


I89 


EXPERIMENT  S2.  Ohm's  method  for  the  measurement  of  the 
E.  M.  F.  of  a  battery. 

The  object  of  this  experiment  is  the  determination  of  an 
E.  M.  F.  in  absolute  measure,  without  reference  to  any  standard 
cell.  From  Ohm's  law  we  have 

E 


1  = 


(123) 


in  which  RQ  is  the  constant  unknown  resistance  of  the  battery, 
connecting  wires,  and  galvanometer ;  and  R  is  the  known 
resistance  which  may  be  varied  at  pleasure. 


40 


30 


£20 

cc 


10 


E.  M.  F 


BY  OHMS  METHOD 


O  10  20  30  40  50  6O  7O 

RESISTANCE 

Fig.  65. 

The  procedure  is  as  follows  : 

(1)  Place  in  the  circuit  of  the  generator  a  resistance  R, 
whose  value  in  ohms  is  known,   and  measure  the   current  in 
amperes   by   means   of    an    ammeter  or   galvanometer   whose 
constant  is  known. 

(2)  Vary  the  known  resistance,  and  measure  the  current  as 
before. 

These  two  observations  will  give  two  equations   between 
which  RQ  may  be  eliminated,  and  E  determined  in  volts. 


190 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


The  best  results  will  be  obtained  if  the  two  resistances  are 
so  chosen  as  to  make  the  two  values  of  the  current  quite  differ- 
ent. As  a  check,  it  is  best  to  repeat  the  observations  with  a 
number  of  different  resistances.  It  is  to  be  observed  that  the 
method  depends  upon  the  assumption  that  the  E.  M.  F.  is  unaf- 
fected by  changes  in  the  current.  With  some  cells  this  is  only 
approximately  true. 

When  a  number  of  observations  have  been  taken,  the  results 
can  be  readily  computed  by  graphical  methods.  To  accomplish 
this,  a  curve  should  be  platted  with  resistances  as  abscissas,  and 
reciprocals  of  currents  as  ordinates.  If  the  E.  M.  F.  remains 
constant,  the  curve  obtained  should  be  a  straight  line,  and  from 
the  "  pitch  "  of  this  line,  E  can  be  computed.  The  following 
table  presents  typical  data,  the  results  of  which  are  shown 
graphically  in  Fig.  65  : 

E.  M.  F.  BY  OHM'S  METHOD.  —  Two  GRAVITY  CELLS  IN  SERIES. 


GALVANOMETER 

Box 

READINGS.  . 

DOUBLE 

RESIST- 

T~\.,rT -,„ 

.                     tj 

2  5 

<j 

f~* 

I 

EM    F 

ANCE. 

TION. 

lan  o 

CURRENT 

Current' 

.  JM  .  r  . 

Right. 

Left. 

7 

95.70 

49.04 

46.66 

0.4666 

25°     i' 

0.2218 

0.1686 

5-93 

2.14 

10 

90.66 

53-88 

36.78 

0.3678 

20°  II' 

o.  1  780 

o.i353 

7-39 

2.12 

20 

83.15 

61.28 

21.87 

0.2187 

1  2°  20' 

0.1080 

0.0821 

12.18 

*  2.  II 

30 

79.92 

64.47 

15-45 

0.1545 

8°  47' 

0.0767 

0.0583 

17.15 

2.08 

40 

78.19 

66.09 

I2.IO 

O.I2IO 

6°  54' 

0.0603 

0.0458 

21.84 

2.09 

50 

77.11 

67«I3 

9.98 

0.0998 

5°  42' 

0.0498 

0.0384 

26.05 

2.14 

60 

76.35 

67.85 

8.50 

0.0850 

4°  52' 

0.0425 

0.0323 

30.96 

2.12 

70 

75.90 

68.42 

748 

0.0748 

4°  17' 

0.0374 

0.0284 

35.22 

2.15 

Distance  of  mirror  from  scale  =  50  scale  divisions. 

Galvanometer  constant  70  =  0.76  ampere. 

From  curve  J?Q  =  5.7  ohms,   E  —  2.14  volts. 

Last  column  computed  assuming  value  of  RQ  obtained  from  curve. 


POTENTIAL   AND    ELECTROMOTIVE   FORCE.  I9I 

Addenda  to  the  report: 

(1)  Show  that  the  best  results  will  be  obtained  when  the 
two  currents  vary  widely  in  amount. 

(2)  If    the    experiment    were    performed    using    cells    the 
E.  M.  F.  of   which   falls   off   as   the  current   increases,   what 
would  be  the  form  of  the  curve  ? 

EXPERIMENT  S3.     The  potential  difference   at  the  terminals 
of  a  battery  considered  as  a  function  of  the  external  resistance. 

The  difference  of  potential  between  the  terminals  of  a  cell 
has  its  greatest  value  when  the  external  resistance  is  infinite 


Fig.  66- 

(when  the  circuit  is  broken),  and  is  then  equal  to  the  electro- 
motive force.  As  the  external  resistance  is  diminished,  the 
E.  M.  F.  remains  constant ;  but  the  difference  of  potential 
between  the  poles  steadily  grows  less,  until  the  external  resist- 
ance is  zero,  when  the  two  poles  are  at  the  same  potential. 

The  relation  between  these  two  quantities  may  be  investi- 
gated as  follows  : 

(1)  Complete  the  circuit  of  the  battery  by  a  resistance  box. 

(2)  Connect   a   high    resistance    galvanometer    through   a 
reversing   key  to   the  terminals  of   the  resistance  box.     (See 
Fig.    66.)      The   resistance  of   the   galvanometer   used   should 
be    so   great   (1000  ohms  or  more)  that   the   current  passing 
through  it  is  too  small  to  modify  appreciably  the  current  in  the 
main  circuit.      Under  these   circumstances,   the   galvanometer 
merely  serves  to  measure  the  difference  of  potential  between 
the  terminals  of  the  cell. 


192 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


Let  Ig  be  the  current  flowing  in  the  galvanometer,  Rg  its 
resistance,  and  pd  the  potential  difference  between  its  terminals  ; 
then  we  have 


The  product  RgIQ  is  a  constant  for  which  may  be  substituted 
the  symbol  pdQ.  This  is  the  constant  of  the  instrument  used  as 
a  potential  galvanometer, 

.  •.  pd  =  Rgf0  tan  B  =  pd^  tan  8.  (  1  24) 

When  Rg  is  very  great,  this  will  be  very  nearly  the  potential 
difference  that  would  exist  if  no  galvanometer  were  used.  It 


2O  25  3O  35  4O 

RESISTANCE 

Fig.  67. 

may  be  here  nT)ted  that  the  potential  difference  between  the 
terminals  of  the  galvanometer,  the  battery,  and  the  resistance 
box  are  practically  identical,  since  the  connecting  wires  are 
supposed  to  have  negligible  resistance. 

(3)  Observe  the  reading  of  the  galvanometer  for  a  number 
of  different  resistances  in  the  box,  and  also  when  the  circuit 
through  the  box  is  broken.  These  resistances  should  be  such 


POTENTIAL  AND  ELECTROMOTIVE  FORCE. 


193 


that  the  galvanometer  deflections  vary  by  approximately  equal 
steps. 

If  /  is  the  current  in  the  resistance  box,  R  its  resistance, 
and  RI  the  resistance  of  the  cell,  including  the  connecting 
wires  to  the  box,,  we  shall  have 


or 


=ER. 


(125) 


If  the  observations  taken  be  platted  with  box  resistances  as 
abscissas,  and  potential  differences  as  ordinates,  the  resulting 
curve  should  be  a  hyperbola,  with  an  asymptote  parallel  to  the 
axis  of  abscissas. 

Results  obtained  by  the  method  just  described  are  given 
in  the  following  table.  The  relation  between  resistance  and 
potential  difference  is  shown  graphically  in  Fig.  67. 

POTENTIAL  DIFFERENCE  BETWEEN  TERMINALS  OF  GRAVITY  CELL. 


RESISTANCE  IN 
Box. 

GALVANOMETER  READINGS. 

GALVANOMETER 
DEFLECTION 
PROPORTIONAL  TO 

POTENTIAL 
DIFFERENCE  IN 

Direct. 

Reversed. 

pd. 

VOLTS. 

00 

65.26 

9.80 

55-46 

1.065 

2OO 

63.40 

11.70 

5I-70 

0.991 

80 

6  1.  06 

14.00 

47.06 

0.904 

40 

58.06 

17.07 

40.99 

0.787 

25 

55-30 

19.87 

35-43 

0.680 

15 

51.90 

23.30 

28.60 

0-549 

IO 

49.25 

26.IO 

23-15 

0-444 

6 

45-95 

29-35 

16.60 

0.319 

4 

43.90 

3L50 

12.40 

0.227 

2 

41.20 

34-28 

6.92 

0.133 

0.8 

39.20 

36.20 

3-oo 

0.057 

Constant  per  scale  division  of  galvanometer  used  as  potential  instrument  = 

pdQ  =  192  x  io~*. 
From  plat.  J?b  =  13.6  ohms. 


„- 


194  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

If  the  constant  pdQ  is  not  known,  any  quantity  that  is  pro- 
portional to  the  potential  difference  may  be  substituted  for  it  in 
platting  the  curve. 

If  the  observations  be  platted  with  reciprocals  of  box  resist- 
ances as  abscissas,  and  cotangents  of  galvanometer  deflections 
as  ordinates,  the  curve  should  be  very  nearly  a  straight  line, 
whose  intercept  on  the  axis  of  abscissas  is  equal  to  -  —  •  Rt 

R* 

may  also  be  obtained  from  the  first  curve.  It  is  the  abscissa 
corresponding  to  the  ordinate  which  is  half  the  maximum 
ordinate. 

Addenda  to  the  report: 

(1)  From  the  curve  platted,  determine  the  external  resist- 
ance so  that  the  terminal  potential  difference  shall  be  J,  ^,  -^ 
of  the  E.  M.  F.  of  the  battery . 

(2)  Compute  the  resistance  of  the  battery. 

EXPERIMENT  S4.  Principle  of  fall  of  potential  in  a  wire 
carrying  a  current. 

This  experiment  is  intended  to  illustrate  the  fact  that  the 
difference  in  potential  between  any  two  points  on  a  simple  cir- 
cuit in  which  a  current  is  flowing  is  proportional  to  the  resist- 
ance between  these  points.  This  proportionality  of  fall  of 
potential  and  resistance  holds  true  in  the  case  of  any  simple 
circuit,  provided  that  there  is  no  electromotive  force  between 
the  two  points  considered.  It  is  a  direct  consequence  of  Ohm's 
law,  and  may  be  stated  as  follows  : 

pd=Ir,  (126) 

in  which  pd  is  the  difference  of  potential,  r  the  resistance 
between  any  two  points  of  a  simple  circuit,  and  /  the  current 
flowing. 

The  most  direct  method  of  testing  this  proportionality 
would  undoubtedly  be  to  measure  the  difference  of  potential 
between  selected  points  of  a  circuit  by  means  of  an  electrometer. 


POTENTIAL   AND    ELECTROMOTIVE   FORCE. 


195 


In  this  case  the  measurement  would  depend  upon  electrostatic 
forces,  and  the  current  flowing  in  the  circuit  would  not  be 
modified.  The  following  method  will,  however,  give  results 
that  are  quite  closely  correct  if  the  galvanometer  resistance  is 
sufficiently  large. 

The  procedure  is  as  follows  : 

(1)  Connect  a  resistance  box  in  series  with  a  gravity  battery 
of  one  or  more  cells,  and  take  out  all  the  plugs  corresponding 
to  the  low  resistances. 

(2)  Connect  a  high  resistance  galvanometer  through  a  revers- 
ing key  s  to  side  plugs,  and  by  this  means  put  the  galvanometer 


Fig.  68. 


in  multiple  with  a  portion  of  the  resistance  in  the  box  (Fig  68). 
In  order  that  the  galvanometer  shall  not  perceptibly  alter  the 
fall  of  potential  in  the  main  circuit,  the  resistance  between  the 
points  to  which  it  is  connected  should  not  be  greater  than  0.005 
that  of  the  galvanometer.  If  the  galvanometer  deflection  is  not 
a  suitable  one,  it  should  be  made  so  by  varying  the  resistance 
or  E.  M.  F.  in  the  main  circuit. 

The  side  plugs  should  now  be  shifted  from  place  to  place  on 
the  box  so  as  to  include  different  resistances  between  them  ; 
and  for  each  value  of  the  included  resistance  the  deflection  of 
the  galvanometer  (both  direct  and  reversed)  should  be  observed. 
Ten  or  twelve  different  values  of  the  resistance  included  between 
the  plugs  should  be  used,  ranging  from  the  smallest  that  will 


196  JUNIOR   COURSE    IN    GENERAL   PHYSICS. 

give  a  readable  deflection,  to  the  largest  that  can  be  used  with- 
out throwing  the  galvanometer  reading  off  the  scale. 

Since  the  resistance  of  the  galvanometer  remains  constant 
throughout  the  experiment,  the  current  passing  through  it  is  in 
each  case  proportional  to  the  difference  of  potential  between  the 
side  plugs. 

The  results  may  be  best  used  to  test  the  principle  of  fall  of 
potential  by  platting  a  curve  in  which  resistances  between  side 
plugs  are  used  as  abscissas  and  the  corresponding  galvanometer 
deflection  as  ordinates.  (If  the  galvanometer  is  a  tangent  gal- 
vanometer, tangents  of  deflections  must  be  used ;  if  a  sine  gal- 
vanometer, sine  of  deflection,  etc.).  The  curve  should  be  very 
nearly  a  straight  line,  very  slightly  concave  towards  the  axis  of 
abscissas. 

This  experiment  will  be  even  more  instructive  if  a  straight 
wire  60  or  80  cm.  long  whose  resistance  is  about  0.005  that  of 
the  galvanometer,  be  substituted  for  the  resistance  box  with 
side  plugs.  If  this  is  done,  one  terminal  of  the  galvanometer 
should  be  connected  to  one  end  of  the  wire  and  the  other  ter- 
minal to  a  sliding  contact,  by  means  of  which  the  length  of  wire 
between  the  galvanometer  terminals  may  be  varied  at  pleasure. 
If  the  cross-section  of  the  wire  is  uniform,  it  will  be  found  that 
difference  of  potential  is  proportional  to  the  length  of  wire 
included  between  the  galvanometer  terminals. 

The  principles  involved  in  the  use  of  a  galvanometer  as  a 
voltmeter  will  be  brought  out  quite  clearly  if  a  new  series  of 
observations  is  taken  in  which  the  resistances  between  the  side 
plugs  range  up  to  ^  or  J  of  the  resistance  of  the  galvanometer. 
This  may  be  done  by  very  greatly  increasing  the  resistance  of 
the  main  circuit.  If  in  this  case  the  observations  be  platted  as 
before,  there  will  be  a  very  decided  curvature  toward  the  axis 
of  abscissas ;  but  if  the  true  multiple  resistance  between  the 
side  plugs  be  used  as  abscissas,  the  curve  rigorously  becomes  a 
straight  line. 


POTENTIAL  AND  ELECTROMOTIVE  FORCE. 


I97 


Addenda  to  the  report: 

(1)  Indicate  the  circumstances  under  which  there  would  be 
no  curvature  in  a  series  of  observations  platted  as  above. 

(2)  Why  does  the  curve  become  straight  when  platted  as 
described  in  the  last  paragraph  of  the  directions  ? 

EXPERIMENT  S5.     Beetz's  method  of  measuring  electromotive 
forces. 

This  experiment  depends  upon  rinding  two  points,  A  and  B, 
(Fig.  69)  in  the  circuit  of  the  battery  whose  E.  M.  F.  is  required, 


G 

Fig.  69. 

such  that  their  potential  difference  shall  equal  the  E.  M.  F.  of 
a  standard  cell.  If  r  is  the  resistance  between  these  points, 
R  the  total  resistance  of  the  circuit  exclusive  of  the  battery, 
whose  resistance  is  Rb,  pd  the  fall  of  the  potential  between  A 
and  B  (which  is  equal  to  the  E.  M.  F.  of  the  standard  cell),  the 
current  in  the  principal  circuit  will  be  equal  to 

-p~  (127) 


If  a  new  value  of  R  is  taken,  r  must  also  be  changed.  This  will 
give  another  equation  similar  to  the  above,  and  between  them 
Rb  may  be  eliminated  and  the  ratio  of  E  to  pd  computed. 

The  method  may  also  be  employed  to  determine  the  battery 
resistance  Rb,  but  good  results  cannot  be  expected  unless  R 
is  always  comparable  with  Rb. 


198  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

To  perform  the  experiment : 

(1)  Connect  the  unknown  E.  M.  F.  in  series  with  a  known 
resistance  which  may  be  varied  at  pleasure.     If  the  current  in 
the  main  circuit  flows  from  A  to  B,  connect  the  negative  pole  of 
the  standard  cell  to  B  and  the  positive  pole  to  a  galvanometer. 
The  other  terminal  of  the  galvanometer  is  connected  to  A. 

(2)  Vary  the  position  of  the  point  A  or  B  or  both,  until  no 
current  flows  through  the  galvanometer.     Under  these  circum- 
stances the  fall  of  potential  from  A  to  B  due  to  the  current  in 
the   main    circuit   is   equal   to   the    E.  M.  F.    of  the   standard 
cell. 

The  experiment  consists  in  finding  a  number  of  values  of  R 
and  r  such  that  no  current  flows  through  the  galvanometer.  From 
any  two  pairs  of  values  both  E  and  Rb  may  be  determined.  Or 
better,  plat  the  values  of  R  as  abscissas,  and  those  of  r  as  ordi- 
nates.  This  will  give  a  straight  line  from  whose  constants  E 
and  Rb  may  be  determined. 

The  standard  may  be  a  Daniell  cell  in  the  form  of  a  U-tube 
in  the  bottom  of  which  is  placed  plaster  of  Paris  to  prevent  the 
mixing  of  the  two  solutions,  or  a  Clark  cell.  In  either  case  the 
internal  resistance  will  be  very  large ;  but  if  the  galvanometer 
is  sufficiently  sensitive,  this  fact  has  no  influence  on  the 
results. 

If  the  battery  whose  E.  M.  F.  is  to  be  measured  is  subject 
to  rapid  polarization,  the  current  should  be  allowed  to  flow  only 
for  an  instant  before  closing  the  circuit  of  the  galvanometer. 

By  this  method  it  is  obvious  that  the  E.  M.  F.  to  be  measured 
must  be  greater  than  that  of  the  standard  cell.  Unless  the 
galvanometer  is  very  sensitive,  it  should  be  three  or  four  times 
as  large. 

Addendum  to  the  report : 

Show  that  when  no  current  flows  in  the  galvanometer  the 
potential  difference  between  A  and  B  is  equal  to  the  E.  M.  F.  of 
the  standard  cell. 


POTENTIAL   AND    ELECTROMOTIVE   FORCE.  199 

EXPERIMENT  S6.  To  trace  the  lines  of  equal  potential  in  a 
liquid  conductor. 

The  apparatus  for  this  experiment  consists  of  a  shallow 
vessel  provided  with  a  glass  bottom  and  rilled  with  some  poorly 
conducting  liquid,  such  as  ordinary  water.  A  telephone  is  also 
required,  and  some  means  of  obtaining  an  alternating  or  inter- 
rupted current.  A  small  induction  coil  is  suitable  for  this 
purpose. 

If  two  electrodes  are  placed  in  the  liquid  and  a  current 
passes  between  them,  the  current  will  flow  from  one  electrode 
to  the  other  by  every  possible  path.  The  potential  varies  along 
each  of  these  paths,  having  its  greatest  value  at  the  positive 
pole  and  its  least  value  at  the  negative  pole.  For  each  value 
of  the  potential  between  these  limits,  there  is  therefore  a  point 
on  each  of  these  "lines  of  flow."  Since  all  these  points  are  at 
the  same  potential,  they  lie  upon  one  of  the  equipotential  lines 
of  the  liquid.  The  object  of  this  experiment  is  to  determine 
the  shape  of  these  equipotential  lines. 

Connect  two  wires  to  the  terminals  of  the  telephone,  and 
fasten  one  of  them  so  that  its  end  dips  into  the  liquid.  If  the 
end  of  the  second  wire  is  also  placed  in  the  liquid,  a  sound  will 
in  general  be  heard  in  the  telephone,  due  to  the  rapid  make  and 
break  of  the  current.  By  shifting  the  position  of  the  second 
wire,  however,  a  position  can  be  found  such  that  this  sound  is 
no  longer  heard.  When  this  position  is  reached,  the  ends  of 
the  two  wires  must  be  at  the  same  potential,  and  are  therefore 
points  on  the  same  equipotential  line.  Keeping  the  position  of 
the  first  wire  unaltered  and  varying  that  of  the  second,  enough 
points  can  be  found  in  this  way  to  locate  the  equipotential  line 
with  considerable  accuracy.  These  lines  should  be  traced  quite 
carefully  near  the  edge  of  the  conductor,  and  in  the  neighbor- 
hood of  a  line  separating  a  good  conductor  from  a  bad  con- 
ductor. It  will  be  found  convenient  to  place  a  board  ruled  with 
equidistant  lines  beneath  the  glass  bottom  of  the  vessel,  and  to 
record  the  position  of  the  points  by  reference  to  these  lines. 


200  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

A  diagram  can  afterwards  be  drawn  on  which  the  equipotential 
curves  are  accurately  represented.  To  avoid  annoyance  from 
the  noise  of  the  interrupter  on  the  induction  coil,  it  is  advisable 
to  place  the  latter  in  a  separate  room. 

In  the  manner  described  above,  the  form  of  the  equipoten- 
tial lines  may  be  investigated  when  electrodes  of  different 
shapes  are  used,  or  when  the  relative  position  of  the  electrodes 
is  altered.  In  each  case,  at  least  five  or  six  lines  should  be 
located,  the  intervals  between  them  being  so  chosen  that  the 
field  in  all  parts  of  the  liquid  is  clearly  shown.  Diagrams  should 
be  drawn  to  scale,  representing  the  position  of  the  electrodes 
and  the  limits  of  the  vessel,  as  well  as  the  equipotential  curves. 
Since  the  lines  of  flow  must  be  at  all  points  perpendicular  to 
the  equipotential  lines,  the  former  can  also  be  drawn. 

Very  instructive  results  may  be  obtained  by  placing  between 
the  electrodes  a  piece  of  metal  of  high  conductivity.  Since 
the  resistance  of  the  metal  is  less  than  that  of  the  liquid,  the 
field  will  be  distorted,  and  the  modified  form  of  the  equipoten- 
tial lines  can  be  determined  by  the  telephone.  A  piece  of  some 
poorly  conducting  substance,  such  as  glass  or  paraffin,  will  also 
give  instructive  results. 

It  may  sometimes  be  desirable  to  use  a  galvanometer  in- 
stead of  a  telephone  in  tracing  the  equipotential  lines.  In 
this  case,  a  continuous  instead  of  an  alternating  current  must 
be  used,  and  the  liquid  conductor  may  be  replaced  by  a  sheet 
of  tinfoil.  The  equipotential  lines  are  determined  by  finding 
a  series  of  points,  such  that  if  the  galvanometer  terminals  be 
connected  to  any  two  of  them,  no  current  will  flow  through 
the  galvanometer. 

On  account  of  the  analogy  that  has  been  found  to  exist 
between  lines  of  flow  and  magnetic  lines  of  force,  the  results 
of  this  experiment  have  important  bearings  on  magnetic  prob- 
lems, such  as  occur  in  dynamo  work. 


POTENTIAL   AND   ELECTROMOTIVE   FORCE.  2OI 

Addenda  to  the  report: 

(1)  Indicate   the   reason   why  the   equipotential   lines   are 
always  normal  to  the  edge  of  the  conductor. 

(2)  Indicate  the  part  of  the  conductor  in  which  the  current 
density  is  the  greatest. 

(3)  Indicate  the  part  of  the  conductor  in  which  the  fall 
of  potential  is  most  rapid. 

EXPERIMENT  S7.  Variation  in  the  E.  M.  F.  of  a  thermo- 
element with  change  in  temperature. 

There  is  always  a  difference  of  potential  between  points 
on  opposite  sides  of  the  junction  between  two  different  metals. 
If  two  metals  be  joined  so  as  to  make  a  complete  circuit,  there 
will  be  a  fall  of  potential  at  each  junction.  Since  these  two 
changes  of  potential  are  equal  and  are  opposed  to  each  other, 
no  current  will  be  produced.  In  a  word,  the  whole  of  one 
metal  will  be  at  one  potential,  while  the  whole  of  the  other 
metal  will  be  at  a  different  potential. 

This  contact  difference  of  potential  depends  upon  tempera- 
ture. Therefore,  if  the  two  junctions  are  at  different  tempera- 
tures, these  two  differences  of  potential  will  not,  in  general, 
annul  each  other,  and  a  constant  current  will  flow  through  the 
circuit.  Such  a  combination  of  two  metals  with  the  two  junc- 
tions at  different  temperatures  constitutes  a  thermo-element 
It  is  the  seat  of  a  true  E.  M.  F.,  as  that  term  has  already 
been  defined. 

It  is  the  object  of  this  experiment  to  determine  the  relation 
between  this  E.  M.  F.  and  the  temperatures  of  the  two  junc- 
tions. 

The  procedure  is  as  follows  : 

(i)  Construct  a  simple  form  of  element  by  soldering  together 
the  ends  of  two  wires  made  of  different  metals  :  for  example, 
German  silver  and  copper,  or  copper  and  iron.  Then  cut  one 
of  the  wires  in  the  center,  so  that  the  free  ends  will  form 
the  terminals  of  the  element. 


2O2  JUNIOR   COURSE    IN   GENERAL   PHYSICS. 

(2)  Connect    the   terminals  of  the  element  through   a  re- 
versing key  to  a  sensitive  galvanometer.     It  will  be  advisable 
to  place  a  resistance  box  somewhere  in  the  circuit,  so  that  the 
resistance  of   circuit   may  be  under   control.      The  resistance 
of  the  whole  circuit  should  be  great  enough  so  that  it  will  not 
be  appreciably  altered   by  changes  in  the  temperature  of  the 
element. 

(3)  One  junction  of   the   element   is  now  to  be  kept  at  a 
constant  temperature,  while  the  other  is  placed  in  a  bath  of  oil 
or  water  whose  temperature  can  be  readily  varied.    The  E.  M.  F. 
corresponding  to  any  observed    difference    in  temperature  be- 
tween the  junctions  is  then  proportional  to  the  galvanometer 
deflections,  or  its  tangent,  as  the  case  may  be.     It  is  important 
that  the  terminals  of  the  element  be  kept  at  the  same  tempera- 
ture.    This  can  usually  be  accomplished  by  placing  them  side 
by  side  and  wrapping  them  with  paper. 

The  junction  whose  temperature  is  to  be  varied  should  be 
inserted  in  a  test-tube  for  protection  against  the  chemical  action 
of  the  bath.  Its  temperature  may  be  measured  by  a  thermom- 
eter placed  in  the  same  tube,  the  bulb  of  the  thermometer 
being  on  a  level  with  the  junction.  To  prevent  air  currents 
it  is  best  to  fill  the  upper  part  of  the  tube  with  cotton  waste 
or  asbestos.  For  rough  work,  the  other  junction  may  be  left 
in  the  air,  provided  it  is  protected  from  draughts.  It  is  better, 
however,  to  place  the  junction  in  some  constant  temperature 
bath,  such  as  boiling  water,  melting  ice,  or  water  that  is  nearly 
at  the  temperature  of  the  room.  In  this  case  the  junction 
should  be  inserted  in  a  test-tube,  as  described  above. 

Observations  of  temperatures  and  galvanometer  readings 
should  be  taken  throughout  a  considerable  range,  the  differ- 
ences of  temperature  between  the  junctions  varying  from  o°  to 
1 00°  or  greater.  Considerable  difficulty  is  often  experienced 
on  account  of  the  uncertainty  of  the  exact  temperature  at  the 
instant  the  galvanometer  is  read.  To  reduce  this  source  of  error 
as  much  as  possible,  take  several  galvanometer  readings  and 


POTENTIAL  AND  ELECTROMOTIVE  FORCE. 


203 


several  temperature  readings,  while  the  temperature  is  main- 
tained as  nearly  constant  as  possible,  and  use  their  average  in 
the  computations. 

(4)  From  the  galvanometer  constant,  the  resistance  of  the 
circuit,  and  the  galvanometer  deflections,  compute  the  E.  M.  F. 
of  the  thermo-element  for  each  observed  difference  of  temper- 
ature. 

(5)  Plat  a  curve  with  temperature  differences  as  abscissas 
and  E.  M.  F.'s  in  microvolts  as  ordinates. 

The  E.  M.  F.  of  the  thermo-element  may  be  determined  by 
a  method  similar  to  that  used  in  experiment  S5.  For  this  pur- 
pose place  in  the  main  circuit  a  galvanometer  to  measure  the 
current  flowing.  Replace  the  standard  cell  by  the  thermo- 
element, and  adjust  the  points  A  and  B  so  that  no  current  flows 
through  the  sensitive  galvanometer  in  the  branch  circuit.  The 
fall  of  potential  between  A  and  B  can  be  determined  from  the 
resistance  between  those  points  and  the  current  flowing.  This 
difference  of  potential  is  equal  to  the  E.  M.  F.  in  the  branch 
circuit. 


CHAPTER   VIII. 
GROUP  T:    THE  MEASUREMENT  OF  RESISTANCE. 

(T)  General  statements ;  (Tx)  Measurement  of  resistance  by  the 
Wheatstone  bridge ;  (T2)  Measurement  of  resistance  by  the 
method  of  fall  of  potential ;  (T3)  Specific  resistance ; 
(T4)  Determination  of  the  temperature  coefficient  for  resist- 
ance of  carbon  and  of  various  metals ;  (T5)  Measurement 
of  the  internal  resistance  of  a  battery  by  Ohm  s  method; 
(T6)  Resistance  of  a  battery  by  Mances  method ;  (T7)  Re- 
sistance of  electrolytes. 

(T).     General  statements  concerning  resistance. 

When  two  points  of  a  homogeneous  conductor  are  main- 
tained at  different  potentials,  a  current  will  flow  in  the  conductor. 
The  magnitude  of  this  current  depends  upon  the  substance  and 
dimensions  of  the  conductor.  The  conductivity  of  a  conductor 
is  that  quantity  which  must  be  multiplied  into  the  potential 
difference  at  its  terminals  to  give  the  current  which  flows.  The 
resistance  of  a  conductor  is  the  constant  ratio  between  the 
difference  of  potential  at  its  terminals  and  the  current  which 
this  potential  difference  produces. 

The  absolute  unit  of  resistance  is  the  resistance  of  a  conductor 
such  that  unit  electromagnetic  difference  of  potential  at  its  ends 
will  cause  unit  electromagnetic  current  to  flow.  It  may  be 
shown  experimentally  that  the  resistance  of  a  conductor  varies 
directly  as  its  length,  and  inversely  as  its  cross-section.  On 
account  of  the  relative  ease  with  which  a  conductor  of  some 
standard  substance  of  given  length  and  section  may  be  con- 

204 


THE   MEASUREMENT   OF   RESISTANCE. 


205 


structed,  it  is  more  usual  to  define  the  practical  unit  of 
resistance  in  these  terms.  The  Chamber  of  Delegates  at  the 
Chicago  Electrical  Congress  adopted,  "As  a  unit  of  resistance 
the  international  ohm,  which  is  based  upon  the  ohm  equal  to  io& 
units  of  resistance  of  the  C.  G.  S.  system  of  electromagnetic  units, 
and  is  represented  sufficiently  well  by  the  resistance  offered  to  an 
unvarying  electric  current  by  a  column  of  mercury  at  the  tempera- 
ture of  melting  ice,  14.4521  grams  in  mass,  of  a  constant  cross- 
sectional  area,  and  of  a  length  of  106.3  centimeters" 

In  current  electricity  it  is  necessary  to  have  variable  resist- 
ances, such  that  any  known  value  may  be  inserted  in  a  circuit 
at  pleasure.  This  demand  is  met  by  constructing  a  series  of 
coils  of  wire  of  different  resistances  and  enclosing  them  in  a 
box,  the  whole  being  callecl  a  rheostat  or  resistance  box.  These 
coils  are  constructed  of  insulated  wire,  usually  of  German  silver. 
This  metal  is  used  for  two  reasons  :  (i)  its  specific  resistance  is 
considerably  greater  than  that  of  copper  or  iron,  thereby  giving 
the  same  resistance  with  less  length  of  wire ;  (2)  the  change  of 
resistance  with  change  of  temperature  is  much  less  than  in  the 
case  of  any  pure  metal.  These  coils  are  non-inductively  or 
doubly  wound,  so  that  their  self-induction  shall  be  as  small 
as  possible.  The  ends  of  each  coil  are  connected  to  separate 
brass  blocks  which  are  electrically  connected  by  remov- 
able brass  plugs.  When  all  of  these  plugs  are  in  place,  the 
resistance  between  the  binding-screws  of  the  box  is  inappreci- 
able. Any  desired  resistance  may  be  introduced  into  the  circuit 
by  removing  the  plugs  corresponding  to  the  proper  resistance 
coils. 

In  the  use  of  resistance  boxes  it  should  always  be  remem- 
bered that  the  resistance  apparently  in  circuit  is  not  the  true 
resistance  unless  each  plug  in  place  makes  good  connection 
between  the  adjacent  brass  blocks.  If  the  plug  be  simply 
dropped  into  place,  or  if  it  be  not  thoroughly  clean,  the  resist- 
ance between  it  and  either  brass  plug,  instead  of  being  infini- 
tesimal, may  have  a  large  value.  Unless  care  is  taken, 


206 


JUNIOR   COURSE   IN    GENERAL  PHYSICS. 


unknown  resistance  thus  introduced  into  the  circuit  is  likely 
to  be  a  considerable  fraction  of  an  ohm.  If  the  resistances 
used  are  small,  this  becomes  of  great  relative  importance.  To 
avoid  this  difficulty,  each  plug  when  it  is  inserted  should  be 
twisted  in  its  seat,  thus  securing  good  contact.  Sometimes 
it  is  necessary  to  clean  the  plugs  and  brass  blocks  with 
emery  paper. 

The  coils  of  resistance  boxes  are  generally  wound  with 
small  wire,  hence  they  should  only  be  used  for  weak  currents. 

EXPERIMENT  Tr  Measurement  of  resistance  by  the 
Wheatstone  bridge. 

By  far  the  most  accurate  method  of  measuring  resistance 
is  by  means  of  the  Wheatstone's  bridge. 

Let  ABC  and  AB'C  (Fig.  70)  be  the  two  parts  of  a  divided 
circuit  containing  no  E.  M.  F.  If  by  means  of  a  battery  a 


Fig.  70. 

current  is  made  to  flow  from  A  to  C,  the  potential  will  fall  from 
A  to  C  along  both  branches.  Let  B  and  Bl  be  points  in  the 
two  branches  having  the  same  potential.  Let  pd  and  pd'  be 
the  differences  of  potential  between  A  and  B  and  between 
B  and  C  respectively.  Let  the  resistances  be  r-±  and  x.  As. 
no  current  can  flow  through  the  branch  BB\  we  have  from 

Ohm's  law  : 

pd__  pd' 


THE   MEASUREMENT   OF   RESISTANCE.  2O/ 

As  B  and  B'  are  at  the  same  potential,  the  difference  of 
potential  between  A  and  B  must  equal  that  between  A  and  B'  . 
Therefore,  in  the  lower  branch  we  have 


whence  x=^R.  (128) 

A  Wheatstone  bridge  is  an  apparatus  consisting  of  three 
sets  of  wire  coils  whose  resistances  are  known.  R  is  a 
rheostat  or  variable  resistance  in  which  any  resistance  may 
be  obtained  from  o.i  to  10,000  ohms,  r^  and  r2  are  called 
"  ratio  arms,"  each  consists  of  a  series  of  resistances,  which 
may  be  made  i,  10,  100,  or  1000  ohms  at  pleasure. 

To  measure  a  resistance  with  this  apparatus,  connect  the 
three  sets  of  resistance  coils,  rly  r2,  and  R,  the  unknown  resist- 
ance, a  sensitive  galvanometer,  and  a  battery,  as  in  the  diagram. 
By  removing  plugs,  make  r^  :  r2  any  convenient  ratio,  say 
10  :  100.  Vary  the  resistance  in  the  rheostat  until  no  current 
flows  through  the  galvanometer  connected  between  B  and  B'. 
The  unknown  resistance  may  then  be  computed  from  the 
known  resistances  of  three  of  the  four  branches. 

In  measuring  resistances  with  the  Wheatstone  bridge,  two 
contact  keys  should  be  used  —  one  in  the  battery  branch 
and  one  in  the  galvanometer  branch.  In  order  to  eliminate 
the  effect  of  thermo-currents,  a  reversing  key  should  be  in- 
cluded in  the  battery  branch. 

It  is  essential  for  accurate  results  that  the  battery  key 
should  be  closed  first,  and  held  closed  long  enough  for  the 
current  to  become  steady,  before  the  galvanometer  circuit  is 
completed.  Otherwise  a  deflection  may  be  produced  on  closing 
the  battery  circuit  even  when  the  bridge  is  properly  balanced. 
This  is  due  to  the  fact  that  the  distribution  of  a  current  when 
first  started  is  determined  largely  by  the  relative  values  of 
the  self-induction  in  different  branches  of  the  circuit,  and 


208  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

does  not  depend  solely  on  the  resistances,  as  is  the  case  when 
the  current  has  become  steady.  The  effect  of  disregarding 
this  precaution  when  measuring  inductive  resistances,  such  as 
electromagnets  or  the  field  coils  of  a  dynamo,  is  always  to 
make  the  resistance  appear  larger  than  it  really  is. 

This  fact  may  be  illustrated  by  selecting  as  one  of  the 
resistances  to  be  measured  an  electromagnet  of  rather  high 
self-induction.  After  the  resistance  in  the  rheostat  has  been 
so  adjusted  that  no  current  passes  through  the  galvanometer 
when  the  keys  are  closed  in  the  proper  order,  observe  the 
effect  of  closing  the  keys  in  the  reverse  order.  After  the  gal- 
vanometer needle  has  come  to  rest,  observe  the  effect  of  open- 
ing the  galvanometer  key  while  the  battery  key  remains  closed. 

Wheatstone's  bridge  is  often  made  in  the  form  known  as 
the  slide  wire  bridge.  In  this  pattern  r  is  a  rheostat,  and  the 
branch  AB'  C  is  a  straight  wire,  a  meter  long,  of  uniform  cross- 
section.  At  B'  there  is  a  key  which  makes  contact  with  the 
bare  wire.  This  key  is  moved  along  the  wire  until  a  point 
is  found  having  the  same  potential  as  B.  Since  resistance  is 
proportional  to  length  (assuming  the  wire  to  be  cylindrical 
and  homogeneous),  we  have 


in  which  a  and  b  are  the  lengths  of  the  two  segments  of  AB'  C. 

Addenda  to  the  report: 

(1)  Explain  the  effect  observed  in  measuring  an  inductive 
resistance  if  the  galvanometer  key  is  closed  first. 

(2)  Prove  that  the  battery  and  galvanometer  may  be  inter- 
changed without  affecting  the  balance  of  the  bridge. 

EXPERIMENT  T2.     Measurement  of  resistance  by  the  method 
of  fall  of  potential. 

In  any  part  of  a  simple  circuit  not  containing  an  E.  M.  F., 

we  have,  from  Ohm's  law, 

pd=IR,  (130). 


THE   MEASUREMENT   OF   RESISTANCE. 


209 


in  which  pd  and  R  are  the  difference  of  potential  and  resistance 
between  the  points,  and  7  is  the  current  flowing.  In  any  other 
part  of  the  same  circuit,  we  have 

X  =  /£'-,  (131) 

the  current  being  the  same  in  all  parts  of  the  circuit. 

If  one  of  these  resistances  is  known  (r,  Fig.  71),  and 
the  ratio  of  the  two  differences  of  potential  is  determined,  the 
unknown  resistance  may  be 
readily  calculated.  This  ratio 
may  most  easily  be  determined 
by  means  of  a  potential  gal- 
vanometer (see  Exp.  S4).  In 
order  that  the  fall  of  potential 
to  be  measured  shall  not  be 
perceptibly  lessened,  when  the 
galvanometer  is  connected  to 
the  two  points,  its  resistance 


Fig  71, 


should  be  1000  or  more  times 
as  great  as  the  unknown  re- 
sistance. If  the  galvanometer 
resistance  is  not  large,  the 

unknown    resistance  may  still  be    determined  if  we  know  the 
galvanometer  resistance.     The  relation  is  proved  as  follows  : 

Let  pd  and  pd1  be  the  potential  differences  between  the 
terminals  of  the  galvanometer,  when  connected  in  multiple  with 
the  standard  and  unknown  resistance,  respectively ;  then  we 

have 

RRa      R'R. 


R+R,  R'+R, 


(132) 


This  assumes  that  the  total  current  in  the  main  circuit 
remains  constant  throughout  the  experiment. 

This  method  of  measuring  resistance  is  especially  useful  in 
measur  ^ry  small  resistances.  It  is  commonly  used,  for 


2io  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

example,  in  determining  the  specific  conductivity  of  a  wire  of 
which  only  a  small  sample  is  available.  (See  Exp.  T3).  It  is 
also  a  convenient  method  of  measuring  resistance  in  deter- 
mining the  temperature  coefficient  of  a  wire  (see  Exp.  T4),  or 
when  the  variation  of  resistance  is  used  to  measure  temperature 
changes. 

The  unknown  resistance  x  (Fig.  71)  is  connected  in  series 
with  the  standard,  a  variable  resistance,  and  a  battery  of  constant 
E.  M.  F.  (The  variable  resistance  may  be  a  standard  resist- 
ance box.)  Since  the  unknown  resistance  and  the  standard 
are  in  series,  the  same  current  is  flowing  in  each. 

The  fall  of  potential  is  measured  by  connecting  the  galva- 
nometer, through  a  reversing  key,  first  in  multiple  with  the  un- 
known resistance,  and  then  in  multiple  with  the  standard,  and 
observing  the  deflection  in  each  case.  The  ratio  of  the  two 
deflections  is  then  equal  (very  closely)  to  the  ratio  of  the  two 
resistances. 

In  preparing  for  this  experiment,  the  galvanometer  should 
first  be  put  in  multiple  with  the  unknown  resistance,  and  the 
resistance  or  E.  M.  F.  in  the  main  circuit  varied,  until  the 
deflection  is  a  suitable  one.  The  resistance  in  the  main  circuit 
should  not  be  very  small,  for  under  these  circumstances  the 
battery  is  more  apt  to  polarize,  and  the  current  to  change  dur- 
ing the  experiment. 

To  eliminate  various  errors,  it  is  best  to  have  two  reversing 
keys,  one  in  the  battery  circuit  and  the  other  in  the  galvanom- 
eter circuit.  For  each  position  of  the  key  in  the  main  circuit, 
the  direct  and  reversed  reading  of  the  galvanometer  should  be 
observed.  The  reversal  of  the  main  current  eliminates  errors 
due  to  the  thermo-currents  caused  by  differences  in  temperature 
between  different  portions  of  the  circuit ;  while  the  reversal  of 
the  galvanometer  circuit  eliminates  any  error  that  might  be 
caused  by  a  direct  magnetic  action  of  the  current  in  the 
unknown  resistance  upon  the  galvanometer  needle.  In  order 
to  be  sure  of  good  contact,  it  is  best  to  make  the  connections  at 


THE   MEASUREMENT   OF   RESISTANCE.  2li 

the  terminals  of  the  unknown  and  standard  resistances  by  means 
of  mercury  cups. 

Several  independent  determinations  should  be  made.  This 
may  be  done  in  either  of  two  ways : 

(1)  Keeping  the  standard  the  same,  vary  the  galvanometer 
deflection,  (a)  by  changing  the  sensitiveness  of  the  galvanom- 
eter,  (b}  by  varying  the  resistance  or  E.  M.  F.    in   the   main 
circuit. 

(2)  Keeping  the  current  in  the  main  circuit  constant,  vary 
the    standard,   by  putting  the  galvanometer  in  multiple   with 
different  coils  of  the  standard  resistance  box. 

In  taking  observations,  it  is  well  to  alternate  between  the 
unknown  and  standard  resistances,  so  as  to  eliminate  the  error 
which  might  be  introduced  by  a  progressive  change  in  the 
conditions. 

Addenda  to  the  report: 

(1)  Explain  by  diagram  the  necessity  of  having  two  revers- 
ing keys. 

(2)  Compute  the  error  introduced  in  your  case  by  using  a 
galvanometer  whose  resistance  was  not  infinite  compared  with 
the  unknown  resistance. 

EXPERIMENT  T3.     Measurement  of  specific  resistance. 

The  specific  resistance  of  a  substance  is  usually  defined  as 
the  resistance  in  absolute  units  of  a  conductor,  i  cm.  long  and 
i  sq.  cm.  in  cross-section.  Specific  resistance  is  sometimes 
defined  in  terms  of  mass  instead  of  volume ;  i.e.  it  is  the 
resistance  of  a  conductor  i  cm.  long  whose  mass  is  i  gram. 

If  the  resistance,  length,  and  cross-section  of  a  wire  be 
measured,  it  is  obvious,  since  resistance  varies  directly  as 
length,  and  inversely  as  cross-section,  that  its  specific  resist- 
ance may  be  readily  calculated.  The  temperature  at  which 
the  resistance  has  been  determined,  should  be  noted  and 
stated. 


212  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

I. 

If  the  sample  furnished  has  a  resistance  of  several  ohms, 
the  resistance  may  be  measured  by  the  method  of  the  Wheat- 
stone  bridge.  The  measurement  should  be  made  with  great 
care,  using  several  different  ratios,  reversing  the  ratio  arms, 
reversing  the  battery  current,  and  taking  every  precaution  to 
make  the  determination  accurate.  The  temperature  of  the 
bridge  coils  as  well  as  that  of  the  wire  whose  resistance  is 
being  determined,  should  be  observed.  From  these  data,  know- 
ing the  temperature  coefficients  of  the  wire  and  the  bridge  coils, 
and  the  temperature  at  which  the  bridge  is  correct,  the  resist- 
ance of  the  wire  at  o°  can  be  computed. 

II. 

If  the  resistance  of  the  sample  to  be  experimented  on  is  one 
ohm  or  less,  it  should  be  measured  by  the  fall  of  potential 
method.  (See  Exp.  T2.) 

In  either  case  the  length  and  diameter  should  be  measured 
with  the  greatest  care.  The  diameter  may  be  directly  measured 
in  a  number  of  places  by  means  of  a  micrometer  wire  gauge  ;  or, 
better,  the  mean  cross-section  may  be  indirectly  determined 
from  the  mass,  length,  and  density  of  the  specimen.  The  density 
should  be  determined  by  weighing  in  water. 

Addenda  to  the  report: 

(1)  Calculate  the  volume  specific  resistance. 

(2)  Compute  the  specific  resistance  in  terms  of  mass. 

(3)  Compute   the   relative   conductivity,    assuming   that   of 
copper  to  be  100. 

EXPERIMENT  T4.  Determination  of  the  temperature  coeffi- 
cient for  resistance  of  carbon  and  of  various  metals. 

The  resistance  of  all  conductors  varies  with  the  temperature> 


THE   MEASUREMENT   OF   RESISTANCE.  213 

and  the  temperature  coefficient  for  resistance  is  defined  by  the 
equation  •  (133) 


in  which  Rt  and  R0  are  the  resistances  at  temperatures  t°  and 
o°,  respectively,  and  a  is  the  coefficient.  For  metals  a  is 
positive. 

Method  of  the  Wheatstone  bridge. 

The  wire  to  be  tested  should  be  insulated,  and  coiled  in  the 
form  of  a  solenoid  sufficiently  small  to  slip  into  a  long  test  tube, 
or  ordinary  glass  tube  sealed  at  one  end. 

Heavy  insulated  copper  wires  are  to  be  soldered  to  the 
two  ends  of  the  coil  and  brought  to  the  terminals  of  the 
Wheatstone's  bridge.  Make  the  test  wire  of  such  size  and 
length  that  its  resistance  will  be  from  two  to  ten  ohms  (the 
higher,  the  better).  Place  a  thermometer  in  the  tube  with  the 
wire,  the  bulb  being  at  the  center  of  the  coil,  and  fill  the  upper 
part  of  the  tube  with  waste  cotton,  asbestos,  or  similar  material 
to  prevent  the  circulation  of  air.  Then  immerse  the  tube  in  a 
water  bath  and  measure  its  resistance  at  different  temperatures, 
ranging  from  zero  to  the  boiling-point.  Readings  of  resist- 
ance should  be  taken  both  for  increasing  and  decreasing  tem- 
peratures, and  the  thermometer  should  be  read  before  and  after 
each  measurement,  the  mean  of  the  two  readings  being  used. 
Let  the  changes  of  temperature  take  place  very  gradually,  and 
keep  the  water  thoroughly  stirred. 

For  practice  determinations  of  the  temperature  coefficient 
of  carbon,  an  incandescent  lamp  may  be  used  instead  of  a  wire. 

From  the  results  obtained,  plat  a  curve  on  cross-section 
paper,  using  temperatures  as  abscissas,  and  resistances  as  ordi- 
nates.  This  curve,  in  the  case  of  most  metals,  will  be  very 
nearly  a  straight  line.  Draw  a  straight  line  as  nearly  as  possi- 
ble through  all  the  points,  and  determine  its  equation.  From 
this  equation  determine  the  temperature  coefficient  a  and  the 
resistance  at  o°. 


214  JUNIOR   COURSE   IN   GENERAL  PHYSICS. 

II. 

Fall  of  potential  method.     (See  Exp.  T2.) 

In  this  case,  a  wire  of  low  resistance  can  be  used  to  advan- 
tage. Two  wires,  not  necessarily  large,  are  to  be  soldered  to 
each  end  of  the  test  wire,  one  pair  serving  to  carry  the  current, 
and  the  other  pair  leading  to  the  galvanometer. 

When  the  coil  is  at  the  temperature  of  the  room,  adjust  the 
resistance  in  series  with  it,  so  that  the  galvanometer  deflection 
is  about  two-thirds  the  distance  across  the  scale.  It  is  very 
important  that  the  temperature  remain  very  nearly  constant 
during  an  observation  of  the  galvanometer  reading.  The  tem- 
perature should  be  taken  as  nearly  as  may  be  at  the  same 
instant  the  galvanometer  reading  is  observed,  both  direct  and 
reversed.  The  mean  of  these  two  temperature  observations  is 
to  be  used  in  the  computations. 

For  the  determination  of  the  temperature  coefficient  it  is 
not  necessary  to  have  any  absolute  standard  of  resistance.  Since 
galvanometer  deflections  are  proportional  to  resistance,  we  may 
substitute  for  Rt  and  RQ  the  deflections  8t  and  S0  (equation  133), 
or  their  tangents,  if  a  tangent  galvanometer  is  used. 

After  making  the  necessary  readings,  a  curve  should  be 
platted,  with  temperatures  as  abscissas  and  galvanometer 
deflections  as  ordinates.  The  equation  of  this  line  is  then 
to  be  determined,  and  from  its  constants  the  temperature 
coefficient  and  deflection  for  o°  are  to  be  calculated. 

Addenda  to  the  report: 

(1)  Justify  the  substitution  of   galvanometer  deflection  for 
resistances  in  the  above  equation. 

(2)  Using  the  coefficient  determined,  calculate  the  resistance 
at  absolute  zero  of  wire  whose  resistance  is  100  ohms  at  o°  C. 

EXPERIMENT  T6.  Measurement  of  the  internal  resistance  of 
a  battery  by  Ohm's  method. 

This  experiment  requires  the  same  observations  as  Exp.  S2, 
and  the  battery  resistance  may  be  calculated  from  the  observa- 


THE    MEASUREMENT   OF   RESISTANCE.  215 

tions  taken  in  that  experiment,  provided  the  resistance  of  the 
galvanometer  and  of  the  connecting  wires  is  known.     It  is  not 
necessary,  however,  to  know  the  constant  of  the  galvanometer. 
From  Ohm's  law  we  have 


7  = 


in  which  R  is  a  known  resistance,  R*  and  Rg  the  battery  and 
galvanometer  resistances  respectively.  The  last  named  includes 
the  resistance  of  the  connecting  wires.  For  /  may  be  substi- 
tuted 70S  (or  70  tan  8,  in  case  the  current  is  proportional  to  the 
tangent  of  the  deflection  of  the  galvanometer  needle). 

If  two  different  values  of  R  be  taken,  and  the  corresponding 
galvanometer  deflections  observed,  we  shall  have  two  equations 
similar  to  134.  If  one  of  these  equations  be  divided  by  the 
other,  both  E  and  70  will  be  eliminated,  and  Rb  will  be  a 
function  of  known  quantities. 

This  experiment  furnishes  an  excellent  example  of  the 
general  principles  discussed  on  page  4.  The  precautions 
there  suggested  should  be  followed  here  ;  that  is  to  say,  the 
difference  between  the  two  currents  in  the  observations  by 
means  of  which  E  and  70  are  eliminated,  and  Rb  is  determined, 
should  not  be  far  from  the  value  of  the  smaller  one.  Further- 
more, the  resistances  used  should  be  comparable  in  magnitude 
with  the  battery  resistance.  In  order  to  meet  these  conditions 
it  will  be  necessary  to  use  a  non-sensitive  galvanometer  of  low 
resistance,  or  to  adjust  a  sensitive  galvanometer  with  a  shunt  of 
proper  resistance  placed  across  its  terminals. 

The  procedure  is  as  follows  : 

(1)  Connect  the  battery  in  series  with  a  resistance  box,  the 
galvanometer,  and  a  reversing  key. 

(2)  Observe   the   galvanometer  readings   for   eight   or   ten 
different  resistances.     These  readings  should  be  taken  several 
times  for  each  resistance  used,  and  the  mean  deflection  derived 
from  them  should  be  utilized  in  the  computations. 


2i6  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

(3)  From  each  suitable  pair  of  observations  compute  the 
resistance  of  the  battery. 

It  will  be  found  instructive  to  determine  the  resistance  from 
the  observations  graphically,  and  if  a  considerable  number  of 
observations  are  taken,  that  will  usually  be  the  least  laborious 
method  and  sufficiently  accurate. 

To  do  this,  plat  a  curve  with  known  resistances  as  abscissas, 
and  reciprocals  of  currents,  or  of  galvanometer  deflections  as 
ordinates.  The  intercept  on  the  axis  of  abscissas  will  be  the 
resistance  of  the  circuit  outside  of  the  resistance  box. 

The  resistance  of  a  cell  is  sometimes  determined  by  con- 
necting two  cells,  first  in  series,  and  then  in  multiple,  and 
observing  the  galvanometer  deflections  in  each  case.  Between 
two  equations  representing  these  observations,  E  and  70  may 
then  be  eliminated,  and  R0  may  be  determined.  This  method 
assumes  that  the  E.  M.  F.'s  and  internal  resistances  of  the 
two  cells  are  identical.  To  test  this  assumption,  connect  the 
two  cells  in  series,  but  so  that  their  E.  M.  F.'s  are  opposed. 
If  no  current  flows,  their  E.  M.  F.'s  are  equal.  Next  connect 
each  cell  in  turn  in  series  with  the  galvanometer  and  the  same 
low  resistance.  If  the  currents  are  equal,  their  internal  resist- 
ances are  equal  (provided  their  E.  M.  F.'s  are  equal). 

The  following  modification  of  the  method,  which  is  especially 
useful  when  the  law  of  the  galvanometer  is  unknown,  may  be 
used  to  check  the  results  : 

(1)  Connect  the   two    cells   in   multiple,    and    observe   the 
deflection  produced  when  some  known  resistance   is    used   in 
the  box. 

(2)  Join   the  cells  in  series  and   adjust  the  box  resistance 
until  the  deflection  is  the  same  as  before. 

(3)  From  the  values  of   the  two  box   resistances   and  the 
galvanometer   resistance,  compute    the    resistance  of   a  single 
cell. 

The  method  of  this  experiment  is  not  applicable  to  batteries 
that  suffer  marked  variation  from  polarization. 


THE   MEASUREMENT   OF   RESISTANCE. 

EXPERIMENT  T6.  Resistance  of  a  battery  by  Mance's 
method. 

It  is  a  rather  difficult  matter  to  secure  a  satisfactory 
measurement  of  the  internal  resistance  of  a  battery.  Mance's 
method  is  perhaps  the  most  accurate  for  a  battery  that  is 
not  subject  to  rapid  polarization. 

The  battery  whose  resistance  is  required  is  made  one  arm 
of  a  Wheatstone  bridge,  the  other  three  arms  being  adjust- 
able resistances  of  known  value.  The  battery  usually  employed 
with  the  bridge  is  removed  and  replaced  by  a  wire,  the  battery 
key  being  retained  in  its  old  place.  In  this  use  of  the  bridge, 
a  current  flows  through  the  galvanometer  at  all  times,  and  it 
will  be  found  advantageous  to  keep  the  galvanometer  key  closed. 

The  measurement  now  consists  in  so  adjusting  the  resist- 
ances in  the  bridge  that  the  opening  or  closing  of  the  battery  key 
has  no  effect  upon  the  deflection  of  the  galvanometer  needle. 
When  this  adjustment  has  been  obtained,  the  resistance  of 
the  cell  can  be  computed  from  the  ordinary  law  of  the  bridge. 

If  the  deflection  of  the  galvanometer  is  too  great  to  be 
read  on  the  scale,  a  permanent  magnet  may  be  used  to  bring 
the  needle  back.  This  magnet  should  be  kept  as  far  away 
as  is  possible,  however,  in  order  not  to  diminish  the  sensitive- 
ness of  the  galvanometer.  Judgment  must  be  used  in  the 
choice  of  the  resistances  placed  in  the  various  arms,  so  as 
to  secure  the  greatest  sensitiveness  and  at  the  same  time 
as  little  inconvenience  as  possible  from  large  and  variable 
deflections.  If  the  resistance  of  the  battery  is  not  very  great 
(thousands  of  ohms),  it  will  be  best  to  adjust  the  resistance 
of  the  three  arms  of  the  bridge,  so  that  the  greatest  resistance 
is  in  series  with  the  battery  and  galvanometer. 

If  the  battery  polarizes,  even  very  slowly,  there  will  be 
a  drift  of  galvanometer  reading.  This  change  of  the  current 
through  the  galvanometer  must,  of  course,  be  disregarded. 
Sometimes  the  observations  are  still  further  complicated  by 
the  existence  of  some  small  self-induction  in  the  bridge  coils. 


2i8  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

The  effect  of  this  is  to  give  the  galvanometer  needle  a  slight 
inductive  throw,  even  though  the  conjugate  condition  of  the 
four  arms  of  the  bridge  has  been  reached. 

Addenda  to  the  report : 

(1)  Prove  that   the  ordinary  law  of   the   bridge  holds  for 
Mance's  method. 

(2)  A  dynamo  is  like  a  battery  in  the  fact  that  it  is  the  seat 
of  an   E.  M.  F.,  and  has  internal  resistance.     What  difficulty 
would  be  experienced  in  measuring,  by  this  method,  the  internal 
resistance  of  a  dynamo  while  running  ? 

EXPERIMENT  T7.     Resistance  of  electrolytes. 

When  a  current  is  passed  through  an  electrolyte,  the  electro- 
lyte is  decomposed,  and  a  counter  E.  M.  F.  is  always  set  up. 
Often  there  is  also  an  evolution  of  gas  at  one  or  both  ends 
of  the  electrodes.  These  effects  complicate  the  experimental 
determination  of  electrolytic  resistance,  but  the  difficulties 
which  they  introduce  may  be,  in  great  part,  avoided  by  the  use 
of  an  alternating  current  of  short  period. 

The  Wheatstone's  bridge  method  of  measuring  resistance 
may  be  adapted  to  the  determination  of  electrolytic  resistance 
as  follows : 

(1)  An  alternating    current    is    supplied   by   replacing   the 
battery  by  the  secondary  circuit  of  an  induction  coil. 

(2)  The  galvanometer  is  replaced  by  some  means  of  detect- 
ing alternating  currents.     A  telephone  will  serve  this  purpose 
very   well.      The   method    of    working    is    analogous    to    that 
described  in  Exp.  Tr      The  resistance  of  the  bridge  arms  is 
varied  until  no  sound  is  heard  in  the  telephone,  and  the  un- 
known resistance   is    determined   by  the  ordinary  law  of  the 
bridge.     Since  the  current  flowing  is  a  rapidly  fluctuating  one, 
it  is  of  the  utmost  importance  that  the  bridge  arms  have  no 
self-induction.      For  this    reason,   a  special   form  of  bridge  is 
generally  used. 


THE   MEASUREMENT   OF   RESISTANCE.  219 

If  the  vessel  containing  the  electrolyte  is  a  tube  or  a  pris- 
matic trough  with  electrodes  filling  the  ends,  the  specific  resist- 
ance may  be  computed  as  in  Exp.  T3.  In  this  way  we  may 
determine  the  specific  resistance  of  different  solutions,  or  of 
the  same  solution  at  different  temperatures  and  densities.  The 
temperature  of  the  solution  should  always  be  noted  at  the  time 
of  the  experiment. 

If  the  vessel  used  does  not  admit  of  accurate  measurement, 
it  should  be  standardized  a  follows : 

(1)  Make  a  10  per  cent  solution  of  zinc  or  copper  sulphate. 

(2)  Fill  the  vessel  and  determine  its  resistance. 

(3)  From  this  resistance  and  the  specific  resistance  of  the 
electrolyte  taken  from  tables,  compute  what  must  be  the  length 
of  the  electrolyte  if  its  cross-section  is  one  square  centimeter. 

The  apparatus  having  been  thus  standardized,  the  specific 
resistance  of  any  other  solution  may  be  determined. 

Before  putting  a  solution  into  the  vessel,  care  should  always 
be  taken  to  scrupulously  clean  the  vessel,  and  to  rinse  it  with 
distilled  water.  The  resistance  of  a  solution  is  sometimes 
greatly  changed  by  even  slight  traces  of  other  substances. 


CHAPTER   IX. 
GROUP  U:    ELECTRICAL  QUANTITY. 

{U)  General  statements  ;  (Uj)  Constant  of  a  ballistic  galvanom- 
eter;  (U2)  Logarithmic  decrement ;  (U3)  Comparison  of 
capacities  ;  (U4)  Capacity  in  absolute  measure. 

(U).     General  statements  concerning  electrical  quantity. 

The  electromagnetic  unit  of  quantity  is  that  quantity  of 
electricity  which  is  transferred  by  unit  current  in  unit  time. 
The  practical  unit  of  quantity,  or  the  coulomb,  is  the  amount 
transferred  by  a  current  of  one  ampere  in  one  second. 

The  total  quantity  of  electricity  transferred  by  any  current 
is  the  product  of  the  current  by  the  time  during  which  it  con- 
tinues. If  the  current  is  variable,  this  becomes 


taken  between  the  proper  limits. 

Quantities  of  electricity  are  considered  when  we  deal  with, 

(1)  The  total  amount  of  an  electrolyte  decomposed. 

(2)  The  charge  and  discharge  of  condensers. 

(3)  Momentary  induced  currents. 

In  cases  2  and  3  the  duration  of  the  current  is  usually  very 
brief,  and  since  the  magnetic  field  produced  is  equally  transient, 
it  is  obvious  that  the  quantity  of  electricity  transferred  cannot 
be  measured  by  means  of  a  galvanometer  used  in  the  ordinary 
manner.  The  quantity  of  electricity  transferred  through  the 
coils  of  a  galvanometer  by  a  momentary  current  can  be  meas- 

220 


ELECTRICAL   QUANTITY. 


221 


ured,  however,  by  the  "  throw  "  or  "  swing  "  of  the  needle  due  to 
the  magnetic  impulse  of  the  momentary  current. 

A  galvanometer  used  for  measuring  such  impulses  is  called 
a  ballistic  galvanometer  from  its  analogy  to  a  ballistic  pendulum. 

Any  galvanometer  can  be  used  as  a  ballistic  galvanometer, 
simply  by  observing  "throws"  instead  of  permanent  deflections, 
provided  that  the  motion  of  the  needle  be  so  slow  that  the  end 
of  the  swing  can  be  determined  accurately.  It  is  also 
desirable,  in  the  case  of  galvanometers  used  ballistically,  that 
the  damping  should  be  as  small  as  possible.  These  two  requi- 
sites are  secured  by  making  the  needle  heavy,  thus  securing 
slow  motion  and  small  factor  of  decrement.  In  using  a  ballistic 
galvanometer,  it  must  be  remembered  that  the  magnetic  moment 
of  the  needle  enters  the  constant  of  the  instrument.  Therefore 
the  needle  should  be  a  magnet  whose  moment  is  not  subject 
to  rapid  change. 

EXPERIMENT  Uj.  Measurement  of  the  constant  of  a  ballistic 
galvanometer. 

There  are  three  methods  for  determining  this  constant : 

(1)  By  measuring  the   throw  of   the   galvanometer   needle 
due  to  the  discharge  of  a  condenser. 

(2)  By  measuring  the  throw  of   the  gal- 
vanometer  needle   produced  by  the  induced 
current  due  to  the  rotation  of   a   coil  in   a 
magnetic  field. 

(3)  By   computation    from    the    periodic 
time   of   the   galvanometer   needle,  and   the 
constant  of  the  instrument  used  as  a  tangent 
galvanometer.     The  last  is  the  most  instruc- 
tive, and  is  the  one  here  given. 

The  amount  of  work  done  against  mag- 
'  netic  forces  in  turning  a  magnet  through  an 
angle  8  (Fig.  72)  in  a  magnetic  field  of  horizontal  intensity  H>  is 

(135) 


222  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

The  kinetic  energy  of  the   magnet   when   it   has   its   greatest 
angular  velocity  is 


The  kinetic  energy  of  the  moving  magnet  at  the  mid-point  is 
equal  to  the  work  necessary  to  turn  the  magnet  through  the 

angle  8. 

(137) 


A  current  of  /  amperes  flowing  in  the  galvanometer  coils  exerts 
a  force  on  the  galvanometer  needle  whose  moment  is  -^MGf, 
G  being  the  true  constant  of  the  galvanometer.  If  the  moment 
of  this  force  be  integrated  over  the  time  during  which  the  cur- 
rent lasts,  it  must  equal  the  moment  of  momentum  produced. 


whence,  -^MGQ  =  K<*^  (138) 


Force  multiplied  by  time  is  equal  to  the  momentum  produced.  In 
the  same  way  moment  of  a  force  multiplied  by  time  is  equal  to  the 
moment  of  momentum  produced  :  but  the  moment  of  momentum  of  a 
body  is  also  equal  to  its  moment  of  inertia  multiplied  by  its  angular 
velocity. 

If   T  is  the  periodic  time  of  the  magnetic   needle  for  small 
oscillations,  we  have  (see  equation  102) 


^im- 

If  in  the  equations  (137),  (138),  and  (139),  K  and  Mbe  expressed 
in  the  form  of  their  ratio,  this  ratio  and  o>0  may  be  eliminated 
between  the  three  equations.  This  will  give 

Q=  io^¥  sin  J5.  (140) 

7T    Lr 
TT 

Now   10—  is  the  working  constant  of  the  galvanometer. 

Gr 

.-.   Q=  r/0sini-8.  (141) 

7T 


ELECTRICAL   QUANTITY.  223 

The  constant  factor  multiplied  into  the  term  sin  \  B  is  the 
constant  of  the  instrument  used  as  a  ballistic  galvanometer. 
Calling  this  quantity  QQ,  we  have 

(2  =  Go  sin}  8.  (142) 

In  order  to  determine  <20,  first  determine  70  as  in  Exp.  Rlt  and 
then  the  periodic  time  of  the  galvanometer  needle  as  in  Exp.  Q3. 
From  these  values  compute  QQ. 

If  8  is  quite  small,  the  quantity  of  electricity  Q  is  propor- 
tional to  8,  but  8  is  proportional  to  the  deflection  on  the  scale. 
Therefore  we  have 


in  which  S  is  the  throw  in  scale  divisions,  and  <20  is  the 
constant  per  scale  division.  The  above  demonstration  assumes 
that  the  whole  of  the  kinetic  energy  of  the  needle  after  the 
current  has  ceased  to  flow  is  used  in  overcoming  magnetic 
forces.  This  is  not  quite  true.  The  friction  of  the  needle 
against  the  air  and  the  current  induced  in  the  galvanometer 
coil  by  the  moving  magnetic  needle,  both  require  the  expendi- 
ture of  energy,  and  therefore  make  S  less  than  it  otherwise 
would  be.  The  theory  of  damping  leads  to  the  conclusion  that 
(i  +  }X)  sin}  8  should  be  substituted  for  sin  }8  in  the  above 
equation,  in  which  X  is  the  logarithmic  decrement  of  the  gal- 
vanometer needle.  (See  Exp.  U2.) 

EXPERIMENT  U2.  Determination  of  the  logarithmic  decre- 
ment of  a  ballistic  galvanometer  needle. 

It  has  already  been  shown  that  the  quantity  of  electricity 
that  passes  through  the  coils  of  the  ballistic  galvanometer  is 
proportional  to  the  impulse  imparted  to  the  needle,  which, 
in  its  turn,  is  proportional  to  the  sine  of  half  the  angle  of  throw, 
or  to  the  angle  itself,  if  the  latter  be  small.  This  is  true,  how- 
ever, only  when  there  is  no  lost  energy  due  to  air  friction  and 
induced  currents,  which  damp  the  oscillation  of  the  needle,  and 
finally  bring  it  to  rest. 


224  JUNIOR   COURSE    IN    GENERAL   PHYSICS. 

Since  it  is  by  means  of  the  throw  that  the  quantity  is  to  be 
measured,  we  must  know  the  correction  that  is  to  be  applied 
to  the  actual  throw  of  the  needle  to  give  the  throw  that  would 
have  resulted  had  there  been  no  damping. 

When  a  magnetic  needle  oscillates  under  the  influence 
of  damping,  the  ratio  of  any  amplitude  to  the  succeeding  one 
in  the  opposite  direction  is  very  nearly  constant,  or 

«i     So      %n 

f=f  =  ^  =  r'  044) 

°2        63        On+l 

This  constant  is  the  "  ratio  of  damping,"  and  its  Napierian 
logarithm  is  called  the  logarithmic  decrement,  and  is  generally 
designated  by  X.  We  have,  therefore, 

*  =  loge-^  (145) 

on+i 

The  equation  of  motion  of  a  body  oscillating  under  the 
action  of  a  force  whose  moment  is  proportional  to  the  angular 
displacement,  as  has  been  shown  under  the  head  of  simple  har- 
monic motion,  is 

K^+G^  =  o.  (146) 

If  the  motion  is  not  simply  harmonic,  but  is  damped  by  friction 
or  otherwise,  a  third  term  must  be  introduced.  In  the  case 
of  an  oscillating  magnet,  damping  is  produced  : 

(1)  By  air  friction. 

(2)  By  induced  currents  due  to  the  motion  of  the  magnet 
near   conductors.       Both   of    these   retarding   forces   are   very 
nearly  proportional  to  the  angular  velocity;  consequently  the 

term  that  must  be  added  to  the  above  equation  is  k  —&-,  in  which 

dt 

k  is  a  constant.  The  complete  equation  of  motion  of  the 
damped  magnetic  needle  is  therefore 


ELECTRICAL   QUANTITY. 
If  we  integrate  this  equation,  we  have 


sn 


225 


(148) 


in  which  S0  is  a  constant,  and  T  is  the  period  of  oscillation  of  the 
needle  under  the  influence  of  damping. 

Let  time  be  reckoned  from  the  instant  the  needle  passes  the 
position  of  equilibrium;   and  let  8V  S2 ,  be  the  values   of 

6  at  the  times  =— ,  - — These  values  of  <f>  will  be  the 

4      4 
successive  actual  amplitudes  of  the  oscillatory  motion  ;  and 


—  one 


SkT 


From  (149)  we  have 


(149) 


(150) 


and  by  substituting  for  this  quantity  X,  as  in  (145),  equation 
149  gives 


Transposing  and  expanding  the  exponential  in  terms  of  X,  and 
neglecting  powers  of  X  higher  than  the  first,  we  obtain 


When  there  is  no  damping,  i.e.  when  k  =  o,  we  have,  from 
(149),  S1  =  S0.  Therefore,  it  follows  that  S0  is  the  quantity 
that  should  be  substituted  for  the  first  actual  throw  in  using 
a  ballistic  galvanometer,  and  that  equation  143  becomes 


The  above  demonstration  is  based  upon  the  assumption 
that  both  S  and  X  are  small.  If  S  is  4°  and  the  ratio  of 
damping  is  1.05,  equation  153  will  be  in  error  by  about  one 


*  See  Gray's  Absolute  Measurements  in  Electricity  and  Magnetism,  vol.  2,  p.  393. 


VOL.  i  —  Q 


. 


226  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

part  in  a  thousand.  If  8  is  10°  and  the  ratio  of  damping  is  1.2, 
the  error  will  be  about  one  in  a  hundred. 

The  object  of  this  experiment  is  to  determine  the  logarith- 
mic decrement  of  a  galvanometer  needle,  and  to  show  the 
relation  of  the  decrement  to  the  resistance  in  circuit  with 
the  galvanometer.  It  is  obvious  that  the  decrement  must 
depend  on  the  resistance,  since  the  damping  is,  in  large  part, 
due  to  the  currents  induced  in  the  galvanometer  coils  by  the 
moving  needle,  and  because  these  currents  are  inversely  pro- 
portional to  the  resistance  of  the  circuit. 

In  the  performance  of  the  experiment,  a  galvanometer 
should  be  used  in  which  the  needle  is  not  strongly  damped. 
From  equation  144,  we  have 

7^-  =  ^>  (154) 

°n+m 

whence  x  =  l  log  —  a-  (155) 

™  °n+m. 

Errors  of  observation  have  the  least  influence  when  the  ratio 
of  Sn  to  Sn+m  is  about  3. 

The  method  of  procedure  is  as  follows  : 

(1)  Set   the   needle   to  vibrating,   and    observe   the   limits 
of  the  successive  swings  to  the  right  and  left  by  means  of  a 
telescope  and  scale. 

(2)  From    these    observations    determine    the     successive 
amplitudes. 

The  position  of  equilibrium  of  the  needle  will  generally 
be  obtained  by  noting  the  scale  reading  when  the  needle  is 
at  rest.  Sometimes  this  position  changes  during  the  progress 
of  an  experiment.  It  may  then  be  obtained  as  follows  :  Let 
Sv  S2,  and  53  be  three  scale  readings  corresponding  to  the 
extremes  of  successive  throws.  We  shall  then  have 


in  which  50  is  the  zero  position  at  the  instant  when  the  scale 


ELECTRICAL   QUANTITY.  22/ 

reading    is    52.      The    deflection    required,    then,    is   in    scale 
divisions, 


If  the  angles  are  not  small,  these  amplitudes  should  be  reduced 
to  circular  measure  by  means  of  the  known  distance  of  the 
scale  from  the  mirror. 

Several  values  of  the  ratio  of  damping  should  be  obtained 
in  the  following  manner:  Suppose  the  («+i)st  amplitude 
to  be  about  one-third  of  the  first  ;  X  should  then  be  determined 
from  the  ratios 


•Wl        °n+2 


Determine  in  this  way  the  logarithmic  decrement  when 
the  galvanometer  coils  are  short-circuited,  and  are  in  open 
circuit,  and  also  for  several  different  resistances,  comparable 
with  the  galvanometer  resistance.  Finally,  from  these  deter- 
minations plat  a  curve,  with  resistances  as  abscissas  and  corre- 
sponding values  of  the  decrement  as  ordinates. 

This  curve  will  have  an  asymptote  parallel  to  the  axis 
of  abscissas,  at  a  distance  from  that  axis  equal  to  the  decrement 
on  open  circuit.  If  the  axis  of  abscissas  be  made  to  coincide 
with  this  asymptote,  the  ordinates  to  the  curve  will  be  the 
decrements  due  solely  to  induced  currents.  These  decrements 
are  inversely  proportional  to  the  resistance  of  the  circuit. 
From  this  relation  and  from  the  curve,  compute  the  resist- 
ance of  the  galvanometer. 

EXPERIMENT  U3.  Comparison  of  the  capacities  of  two 
condensers. 

When  the  coatings  of  a  condenser  are  charged  to  a  potential 
difference,  pd,  the  charge  or  quantity  of  electricity  stored  in 
the  condenser  is 

Q  =  Cpd,  (156) 

in  which  C  is  the  capacity  of  the  condenser.     It  has  already 
been   shown  in   preceding  experiments  that   if  the  quantity  of 


228 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


electricity   Q  is  discharged  through    a   ballistic   galvanometer 
producing  the  deflection  8,  we  have 

If  a  condenser  of  capacity  Cv  charged  to  a  potential  differ- 
ence, pd^  be  discharged  through  the  ballistic  galvanometer, 
we  have 


(158) 


If  another  condenser  of  capacity  C2,  charged  to  a  potential 
difference,  pd^  be  discharged  through  the  same  ballistic  galva- 
nometer, we  shall  have  a  similar  relation.  And  if  the  first 
equation  be  divided  by  the  second,  we  shall  have 

7^  =  4^sT  (J59> 


A  still  simpler  relation  follows  if  the  condensers  have  been 
charged  to  the  same  potential  difference. 

In  experimenting  with  condensers  it  is  generally  necessary 
to  use  rather  large  potential  differences  (from  50  to  several 

hundred  volts).  Such  potential 
differences  may  be  produced  by  a 
water  battery  of  a  sufficient  num- 
ber of  very  small  cells.  Each  cell 
Fi£-73<  consists  of  a  short  test-tube  filled 

with  very  slightly  acidulated  water.  The  plates  are  made  by 
soldering  together,  short  strips  of  copper  and  zinc.  Each 
"couple"  is  bent  into  a  U-shape,  and  the  copper  dipped  into 
one  cell,  and  the  zinc  into  the  next  cell,  as  illustrated  in 

Fig-  73- 

In  condenser  work  it  is  also  necessary  to  use  great  care  in 
securing  good  insulation,  not  solely  on  account  of  the  use  of 
high  potentials,  but  because  the  condenser  must  sometimes 
remain  charged  for  a  few  minutes  while  unconnected  with  a 
battery.  The  procedure  in  this  experiment  is  as  follows  : 


ELECTRICAL   QUANTITY. 


229 


Fig.  74. 


(1)  Connect   the  condenser  in  series  with  the  battery  and 
ballistic  galvanometer,  and  place  in    the  circuit  a  double  con- 
tact key,  as  shown  in  Fig.  74. 

(2)  Make  contact  at  A,  and  thus  charge 
the  condenser   through    the   galvanometer. 
The     corresponding     galvanometer     throw 
should  be  determined  as  in  Exp.  U2. 

(3)  Break  contact  at  A,  and  immediately 
make  contact    at   B,  thus  discharging   the 
condenser  through  the  galvanometer.    The 
galvanometer  needle  will  receive  an  impulse 
in  the  opposite  direction,  which   should  be 
very    nearly    equal    to    the    former    throw. 

These  observations  should  be  repeated  several  times  in  order  to 

get  a  good  average. 

A  similar  series  of  observations  should  now  be  taken  with 
the  condenser  replaced  by  the  one  with  which 
it  is  to  be  compared.  If  the  capacities  of  the 
two  condensers  do  not  differ  greatly,  that  is,  if 
one  is  not  more  than  two  or  three  times  as 
great  as  the  other,  the  same  number  of  cells 
should  be  used.  If  the  difference  of  capacity 
is  very  large,  the  E.  M.  F.'s  of  the  batteries  in 
the  two  cases  should  be  adjusted  to  suit  the 

two  condensers,  and  they  should  be  compared  as  in  Exp.  Sr 
When  condensers  are  connected  as  shown  in  Fig.  75,  they 

are  said  to  be  connected  in  multiple.     If  C   is  the  capacity  in 

multiple,  we  have  the  relation 

I  I  I 


Fig.  75. 


(160) 

This  relation  follows  directly 
from  the  fact  that  the  capacity  of 
a  condenser  is  proportional  to  the 
area  of  either  of  its  coatings. 


Fig.  76. 


When  condensers  are  connected  as  shown  in  Fig.  76,  they 


230  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

are  said  to  be  connected  in  series.     If  C.  is  the  capacity  of  the 
system  in  series,  we  have 

£-£+£+-.  (I6D 

La      Ll       C2 

This  relation  may  be  readily  derived  by  making  use  of  the 
following  facts  : 

(i)  The  potential  difference  at  the  terminals  is  equal  to  the 
sum  of  the  potential  differences  between  the  coatings  of  each 
condenser,  or 

pda  =/</!  +X2  +  •  •  ••  (  1  62) 


(2)  When  several  condensers  are  connected  in  series,  the 
quantity  of  electricity  on  each  coating  of  every  condenser  is 
the  same,  or 

01=0,=-.  (163) 

Equation  161  then  follows  directly  from  the  definition  of 
capacity.  It  is  also  to  be  remembered  that  the  capacity  of  a 
system  of  condensers  connected  in  series  is  the  ratio  of  the 
charge  on  either  extreme  coating  divided  by  the  potential  differ- 
ence between  the  extreme  coatings. 

The  relations  expressed  in  equations  160  and  161  should 
be  verified  experimentally.  This  may  be  done  as  above  by 
comparing  the  series  or  multiple  system  with  a  condenser  whose 
capacity  is  known. 

EXPERIMENT  U4.  Measurement  of  the  capacity  of  a  con- 
denser in  absolute  measure. 

If  the  condenser  is  charged  or  discharged  through  a  ballistic 
galvanometer,  we  shall  have,  as  in  the  preceding  experiment, 

(164) 

If  the  quantities  on  the  right  of  this  equation  are  all  deter- 
mined in  absolute  measure,  the  capacity  will  be  determined 
independently  of  the  capacity  of  any  standard  condenser.  The 


ELECTRICAL   QUANTITY.  231 

constants  QQ,  pd,  and  X  should  be  determined  as  described  in 
previous  experiments.     It  should  be  remembered  that  the  value 
of  X  to  be  used  in  this  experiment  is  that  obtained  when  the 
galvanometer  circuit  is  open. 
The  procedure  is  as  follows : 

(1)  The  throw  of  the  needle  8  is  to  be  determined  as  in 
the  preceding   experiment,  by  charging   and   discharging   the 
condenser  through  the  galvanometer. 

(2)  The  values  of  S,  which  always  differ  in  the  case  of  the 
charge  and  of  the  discharge,  respectively,  should  be  averaged 
separately.     The  former  value  will   correspond  to  the  instan- 
taneous capacity,  while  the  latter  corresponds  to  the  capacity 
of  the  condenser  after  a  greater  or  less  absorption  has  taken 
place. 


CHAPTER    X. 


GROUP  V:    ELECTROMAGNETIC  INDUCTION. 

(V)  General  statements ;  (Vj)  Dip  and  intensity  of  the  earth's 
magnetic  field  (method  of  the  earth  inductor] ;  (V2)  Lines 
of  force  of  a  permanent  magnet ;  (V3)  Mutual  induction. 

(V).     General  statements  concerning  induction. 

Faraday  discovered  that  when   any  portion    of  a  complete 

circuit  is  moved  through  a  magnetic  field,  an  electric  current 

circulates  in  all  parts  of  it. 

This  fact  may  be  viewed  as  follows  : 

Let  there  be  a  conductor,  shown  in  cross-section  (Fig.  77), 

which  forms   part   of   a   complete   circuit.      Suppose  it  to  be 

moving  in  the  direction  of  the  arrow 
in  a  magnetic  field  originally  uniform. 
The  arrangement  of  the  lines  of  force 
of  this  field  is  indicated  by  the  dotted 
lines.  During  the  motion  of  this  con- 
ductor, the  otherwise  uniform  field 
will  be  distorted.  The  field  of  force 
on  the  side  towards  which  the  con- 
ductor is  moving  will  be  stronger 
than  before,  and  the  lines  of  force  will 
be  crowded  together,  and  concave 
towards  the  conductor.  On  the  oppo- 
site side,  the  lines  of  force  will  be 

more   widely    separated,    and    convex   towards    the    conductor. 

Immediately  around  the  conductor,  and  extending  to  a  greater 

232 


Fig.  77. 


ELECTROMAGNETIC   INDUCTION. 


233 


or  less  distance,  according  to  the  intensity  of  the  induced  cur- 
rent, the  lines  of  force  will  be  closed  curves  surrounding  it. 
The  positive  direction  of  the  lines  of  force  in  these  closed 
curves  are  as  indicated  in  the  figure.  It 
follows  that  if  the  direction  of  motion  is  to 
the  right,  and  the  positive  direction  of  the 
lines  of  force  vertically  upward,  the  cur-  . 
rent  will  be  directed  towards  the  observer. 
Or  if  the  motion  is  along  the  ;r-axis,  and 
the  lines  of  force  along  the  ^-axis,  the  cur-  Fig.  78. 

rent  will  be  directed  along  the  j/-axis,  each  in  the  positive  direc- 
tion.    (See  Fig.  78.) 

Since  a  current  may  be  produced  in  this  way,  it  must  be 
that  there  is  an  E.  M.  F.  generated  in  the  moving  conductor. 
This  E.  M.  F.  exists  whether  the  circuit  is  closed  or  not.  In 
the  latter  case,  if  the  motion  is  uniform,  and  in  a  uniform  field, 
there  will  simply  be  a  static  fall  of  potential  along  the  conductor 
in  the  direction  in  which  current  would  flow  if  the  circuit  were 
completed. 

It  has  been  experimentally  demonstrated  that  the  E.  M.  F. 
generated  in  this  way  is  directly  proportional : 

(1)  To  the  velocity  of  motion. 

(2)  To  the  intensity  of  the  magnetic  field. 

(3)  To   the  length    of   the   moving   conductor;    the   three 
directions  being  mutually  perpendicular. 


.D : : : 


H  •  • 


Fig.  79. 

Let  ABCD  (Fig.  79)  be  a  rectangular  circuit  with  one  open 
side,  and  let  it  be  placed  in  a  magnetic  field  of  intensity  Ht  the 
lines  of  force  being  supposed  perpendicular  to  the  plane  of  the 


234 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


paper.  Let  mn  be  a  conductor  resting  on  the  two  parallel  con- 
ductors and  completing  the  circuit.  If  the  length  of  BC  is  /,  and 
mn  moves  in  the  direction  of  the  arrow  with  a  velocity  V=  — , 
the  E.  M.  F.  generated  in  the  circuit  will  be,  in  volts, 


- 

IO8       dt 


(165) 


When  different  parts  of  a  circuit  cut  lines  of  force  at  different 
rates,  the  total  E.  M.  F.  generated  in  the  whole  circuit  is 


IO8" 


Trdx 
J  dt 


(165*) 


J 


It  is  obvious  that  E  may  be  zero  both  when  no  lines  of  force 
are  cut,  also  when  the  E.  M.  F.'s  in  different  parts  of  the  circuit 

due  to  cutting  lines  of  force  are 
oppositely  directed,  and  exactly 
balance  each  other.  For  exam- 
ple, let  AB,  Fig.  80,  be  a  rec- 
tangular circuit  whose  plane  is 
parallel  to  the  lines  of  force, 
and  capable  of  rotation  about 
an  axis  parallel  to  the  lines  of 
force.  Jf  this  circuit  be  rotated 
clock-wise,  E.  M.  F.'s  will  be 
generated  in  the  different  parts, 
as  indicated  by  the  arrows.  These 
obviously  annul  each  other  when  added  around  the  complete 
circuit.  There  is,  however,  an  E.  M.  F.  between  a  and  b  pro- 
ducing a  fall  of  potential  from  a  to  b,  from  a  to  b' ,  from  c  to  d, 
and  from  c  to  d'. 

The  equation  for  the  E.  M.  F.  generated  in  a  complete  circuit 
may  often  be  simplified  in  the  following  manner :  Let  N  be  the 
total  number  of  lines  of  force  that  at  any  instant  pass  through 
the  circuit.  Now  if  the  position  of  the  circuit  is  changed  in  the 


Fig.  80. 


*  For  the  significance  of  the  numerical  factor,  io8,  see  p.  187. 


ELECTROMAGNETIC   INDUCTION.  235 

time  dt  in  such  a  manner  the  change  in  the  number  of  lines  of 
force  that  pass  through  the  circuit  is  dNy  we  have,  for  the  com- 
plete circuit, 


If  the  circuit  is  composed  of  ;/  turns,  through  each  of  which 
the  N  lines  pass,  we  have 


~io8  dt 

Furthermore,  since  Q=j  Idt,  we  have,  for  the  total  quantity  of 
electricity  produced, 

G=^f-  (166) 

in  which  N^  is  the  number  of  lines  of  force  cut,  and  R  is  the 
resistance  of  the  circuit  in  ohms. 

In  making  application  of  the  law  of  induced  E.  M.  F.,  the 
following  fundamental  principles  are  of  service. 

(1)  The  source  of  the  magnetic  field  is  immaterial      It  may 
be  due  to  a  permanent  magnet,  to  the  earth,  or  to  an  electric 
current. 

(2)  It  is  immaterial  whether  the  conductor  is  moved  in  a 
magnetic  field,  or  a  magnetic  field  is  moved  past  the  conductor. 

(3)  Movements  of  the  lines  of  a  magnetic  field  may  be  pro- 
duced in  two  ways  : 

(a)  By  moving  bodily  a  magnet,  or  a  circuit  conveying  a 
current. 

(b)  By  causing  a  current  to  change,  in  which  case  its  lines 
of  force  will  move  outwards  when  the  current  increases,  or  move 
inwards  and  disappear  when  the  current  decreases. 

(4)  An  E.  M.  F.  may  be  induced  in  a  conductor  already  con- 
veying a  current,  and  this  may  either  increase  or  decrease  the 
current  flowing. 

(5)  If  the  current  flowing  in  a  circuit  is  decreased,  the  mag- 
netic field  due  to  the  current  will  decrease,  the  lines  of  force 
collapsing  on  the  conductor.     This  motion  of  the  field  will  pro- 


JUNIOR   COURSE    IN    GENERAL   PHYSICS. 


duce  an  E.  M.  F.  in  the  conductor  tending  to  produce  a  current 
in  the  same  direction  as  the  original  current.  If  the  current  is 
increased,  the  induced  E.  M.  F.  changes  sign.  From  this  it 
follows  that  when  current  is  changed  in  a  circuit,  an  induced 
E.  M.  F.  is  set  up,  which  opposes  that  change.  This  kind  of 
induction  is  called  self-induction. 

EXPERIMENT  Vr  Dip  and  intensity  of  the  earth's  magnetic 
field.  (Method  of  the  earth  inductor.) 

The  earth  inductor  consists  essentially  of  a  coil  of  wire  C, 
Fig.  8 1,  capable  of  revolution  about  an  axis  A,  in  its  own  plane. 

Usually  this  coil  is  mounted  in  a 
frame  F,  which  is  itself  capable  of 
rotation  about  an  axis  A'  in  its  plane, 
perpendicular  to  the  axis  A.  To  this 
frame  is  attached  a  graduated  circle 
»S ;  by  means  of  this  circle  the  angle 
which  the  axis  A  makes  with  a  hori- 
zontal plane  can  be  measured.  The 
instrument  is  also  furnished  with 
stops,  which  enable  the  coil  to  be 
turned  through  exactly  180° ;  and  the 
base  is  furnished  with  leveling  screws,  by  means  of  which  the 
plane  containing  the  two  axes  of  revolution  may  be  made  truly 
horizontal. 

I. 

To  determine  the  angle  of  dip. 

The  angle  of  dip  is  defined  as  the  angle  which  the  direction 
of  the  lines  of  force  makes  with  the  horizon.  If  H  and  V 
are  the  horizontal  and  vertical  components  of  the  intensity 
qf  the  earth's  field  at  any  point,  we  have 


T    T     T 


Fig.  81. 


(167) 


ELECTROMAGNETIC   INDUCTION.  237 

To  determine  /S,  which  is  the  object  of  this  part  of  the 
experiment,  proceed  as  follows  : 

(1)  Turn  the  whole  apparatus  until  the  axis,  about  which 
the  square  frame  revolves,  is   perpendicular  to  the  magnetic 
meridian.     This  may  be  done  with  the  aid  of  a  small  compass. 

(2)  Adjust  the  leveling  screws  until  the  square  frame  con- 
taining^the  two  axes  of  revolution  is  truly  horizontal. 

(3)  Adjust  the  stops  so  that  the  plane  of  the  movable  coil 
is  horizontal  in  both  of  the  extreme  positions. 

When  thus  adjusted,  the  vertical  component  of  the  magnetic 
field  passes  through  the  coil.  In  other  words,  V  lines  of  force 
per  square  centimeter  pass  through  the  coil.  If  n  is  the 
number  of  turns  of  the  coil,  and  A  is  the  mean  area  of  the 
coil,  the  number  of  lines  of  force  passing  through  the  coil  will 
be  nAV. 

(4)  If  the  coil  be  now  turned  through    180°,  all  the  lines 
of  force  will  be  cut  twice,  and  we  have  from  equation  166 

(168) 

in  which  R  is  the  resistance  of  the  circuit.  If  a  ballistic  galva- 
nometer forms  part  of  the  circuit,  and  if  the  coil  be  turned- 
quickly,  we  shall  have 

(169) 

in  which  SF  is  the  throw  of  the  galvanometer  needle.  If  the 
angular  motion  of  the  needle  is  not  small,  sinJS  must  be  used 
instead  of  B. 

(5)  If  the  square  frame  be  now  rotated  through  exactly  90°, 
as  measured  by1  the  divided  circle,  the  number  of  lines  of  force 
passing  through  the  coil  in  its  new  position  will  be  nAH ";  and 
if  Sff  is  the  corresponding  galvanometer  throw,  we  have 


(I/O) 

«&*T 


238  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

The  constants  n,  A,  R,  <20,  and  X  being  the  same  for  both 
positions,  are  eliminated,  and  it  is  not  necessary  to  know  their 
values. 

The  values  Sr  and  SH  used  in  this  computation  should  each 
be  the  mean  of  ten  or  twelve  determinations. 

When  the  square  frame  makes  an  angle  with  the  horizontal 
equal  to  the  dip,  no  lines  of  force  thread  through  the  coil 
in  any  position ;  consequently,  no  current  will  be  produced 
when  it  is  rotated  about  its  own  axis.  The  position  in  which 
no  current  is  produced  by  the  rotation  of  the  coil  should  be 
found  by  trial.  The  angle  through  which  the  frame  was 
turned  from  the  horizontal  position  furnishes  a  second  deter- 
mination of  the  dip. 

II. 

To  determine  intensity. 

From  equation  168  it  is  obvious  that  both  the  vertical  and 
horizontal  intensity  may  be  determined  in  absolute  measure, 
provided  <20  and  X  have  previously  been  determined  for  the 
ballistic  galvanometer,  and  n,  A,  and  R  are  known. 

The  lines  of  force  make  but  a  small  angle  with  the  vertical, 
and  on  this  account  a  small  error  in  leveling  the  coil  will 
produce  a  relatively  great  error  in  the  determination  of  H. 
This  should  be  remembered  in  determining  the  dip  as  well 
as  in  determining  the  horizontal  intensity. 

Addenda  to  the  report: 

(1)  Explain  fully  the  direction  of  the  induced  current  when 
the  coil  is  rotated  about  a  vertical  and  a  horizontal  axis. 

(2)  Explain  from  first  principles  why  there  is  no  current 
when  the  coil  is  rotated  about  an  axis  parallel  to  the  lines 
of  force. 

EXPERIMENT  V2.  Measurement  of  the  lines  of  force  of  a 
permanent  magnet. 

The  object  of  this  experiment  is  the  determination  of  the 
number  of  lines  of  force  that  emerge  from  the  positive  half 


ELECTROMAGNETIC   INDUCTION. 


239 


of  a  permanent  magnet.  Before  beginning  these  measurements 
the  constant  and  the  logarithmic  decrement  of  a  ballistic 
galvanometer  must  have  been  determined.  (Exps.  U1  and  U2.) 
These  values  having  been  ascertained,  the  procedure  is  as 
follows  : 

Connect  a  test  coil  of  a  known  number  of  turns  in  series 
with  the  galvanometer.  Place  the  coil  at  the  center  of 
the  magnet,  and  when  the  galvanometer  needle  has  come  to 
rest  observe  the  throw  of  the  needle  produced  by  quickly 
slipping  the  test  coil  off  the  end  of  the  magnet.  This  test 
coil  should  consist  of  a  considerable  number  of  turns  of  small 
copper  wire,  No.  24-36,  according  to  the  resistance  of  the 


Fig.  82. 


galvanometer  and  the  sensitiveness  of  the  latter.  It  should 
be  of  such  a  form  as  to  fit  easily  over  the  bar  magnet  to 
be  tested.  (See  Fig.  82.) 

If  N  is  the  number  of  lines  of  force  that  emerge  from  the 
magnet,  and  n  the  number  of  turns  in  the  coil,  we  have  from 
equations  153  and  166 


in  which  R  is  the  resistance  of  the  circuit. 

The  most  'suitable  number  of  turns  for  the  test  coil  will 
depend  upon  the  strength  of  the  magnet,  the  sensitiveness 
of  the  galvanometer,  and  the  resistance  of  the  circuit.  These 
quantities  should  be  so  adjusted  that  the  galvanometer  throw 
is  rather  large.  Within  certain  limits  this  result  can  be  most 


240  JUNIOR   COURSE    IN   GENERAL   PHYSICS. 

easily  obtained  by  varying  the  resistance  in  circuit  with  the 
galvanometer. 

If  the  bar  is  not  symmetrically  magnetized,  the  magnetic 
center  must  be  found  experimentally.  To  do  this,  move  the 
test  coil  along  the  bar  step-wise.  When  the  magnetic  center 
is  reached,  a  slight  motion  of  the  coil  in  either  direction  may 
be  made  without  producing  a  reversal  of  current  in  the  galva- 
nometer circuit. 

By  this  method  the  flow  of  induction  from  several  magnets 
should  be  determined,  selecting  for  the  purpose  both  bar 
magnets  and  those  of  the  horseshoe  type. 

Addenda  to  the  report : 

(1)  From  the  readings  obtained  compute  the  induction  per 
square  centimeter  through  the  center  of  each  magnet. 

(2)  If  the  magnetic  moment  is  known,  compute  the  distance 
between  the  poles,  or,   more  properly,   the   distance   between 
the  "  centers  of  gravity  "  of  the  two  distributions  of  magnetism. 

EXPERIMENT  V3.     Mutual  induction. 

The  objects  of  this  experiment  are :  I,  to  observe  certain 
of  the  phenomena  of  mutual  induction ;  II,  to  measure  the 
quantity  of  electricity  which  circulates  in  a  secondary  circuit 
when  the  magnetic  field  in  its  vicinity  produced  by  a  current 
in  a  primary  circuit  is  varied. 

I. 

The  primary  and  secondary  circuits  consist  of  two  coils 
of  the  same  size.  The  primary  coil,  however,  is  wound  with 
considerably  coarser  wire  than  the  secondary  coil. 

The  method  is  as  follows  : 

(i)  Connect  the  primary  coil  P  (Fig.  83)  in  circuit  with 
a  battery  of  constant  E.  M.  F.,  and  insert  a  make  and  break, 
key  K. 


ELECTROMAGNETIC   INDUCTION.  241 

(2)  Connect  the  secondary  coil  5  in  series  with  a  ballistic 
galvanometer  and  a  resistance   box.     The  latter  is  placed   in 
the  circuit  to  enable  the  observer  to  adjust  the  throws  of  the 
galvanometer  needle. 

(3)  Place  the  primary  and  secondary  coils  close   together, 
with  their  axes  coincident. 

(4)  Observe  the  galvanometer  throws  when  the  current  is 
made  and  broken  in  the  primary  circuit. 

(5)  The  circuit  being  closed,  and  current  flowing  steadily 
in  the  primary  circuit,  observe  the   galvanometer  throws  pro- 
duced  by   quickly  moving   the   secondary  to  a  distance   of  a 
meter ;  also  when  the  coil  is  quickly  replaced. 


Fig.  83. 

(6)  Repeat  these  observations,  this   time   moving  the  pri- 
mary instead  of  the  secondary  coil. 

(7)  Observe  the  galvanometer  throw  when  one  of  the  coils 
is  quickly  turned  and  placed  with  its  opposite  face  next  to  the 
other  coil. 

(8)  Observe  the  effect  upon  the  galvanometer  when  a  per- 
manent magnet  is  moved  in  the  vicinity  of  the  secondary  coil. 

II. 

(A)  The  quantity  of  electricity  which  is  produced  in  the 
secondary  circuit  is  directly  proportional  to  the  intensity  of 
the  current  that  is  made  and  broken  in  the  primary  circuit. 
To  prove  this  relationship,  use  the  following  method  : 

VOL.  I  —  R 


242 


JUNIOR   COURSE    IN   GENERAL   PHYSICS. 


(1)  Connect  a  battery,  a  variable  resistance,  and   a  galva- 
nometer in  series  with  the  primary  coil  (Fig.  84). 

(2)  Connect  the   secondary  coil  in   series  with  a  ballistic 
galvanometer,  and  place  the  coil,  as  before,  close  to  the  primary 
with  the  axes  coincident. 

(3)  Observe  the  throws  of  the  ballistic  galvanometer  needle 
when  the  primary  circuit  is  made  and  broken. 

(4)  Measure  also  the  current  flowing  in  the  primary  circuit. 
Repeat  these  observations  with  different    currents    in  the 

primary  circuit.      The  resistances  of   the  two  circuits   should 


Fig.  84. 


be  so  adjusted  that  the  series  of  throws  on  the  ballistic  galva- 
nometer vary  from  the  smallest  that  can  be  accurately  deter- 
mined, to  the  largest  that  the  scale  will  allow.  The  resistance 
of  the  secondary  circuit  must  not  be  changed  during  the  experi- 
ment. 

If  currents  in  the  primary  be  platted  as  abscissas,  and 
throws  of  the  ballistic  galvanometer  as  ordinates,  the  resulting 
curve  will  be  found  to  be  a  straight  line  passing  through  the 
origin.  This  verifies  the  relation 

<2*<*/P,  (172) 

in  which  IP  is  the  current  in  the  primary,  and  Qs  is  the  quantity 
of  electricity  which  circulates  in  the  secondary. 

The  apparatus  described  above  should  be  further  utilized 
to  establish  the  following  relations  : 


ELECTROMAGNETIC   INDUCTION.  243 

(B)  The  quantity  of   electricity  which    is    induced    in  the 
secondary  circuit    is    inversely   proportional   to   the   resistance 
of  that  circuit. 

To  determine  this  fact,  the  same  connections  as  in  (A) 
should  be  made.  Now  observe  the  throws  of  the  ballistic 
galvanometer  when  the  primary  circuit  is  made  and  broken, 
for  several  different  resistances  in  the  secondary  circuit.  If 
a  curve  be  platted  with  resistances  in  the  secondary  circuit 
as  abscissas,  and  the  reciprocals  of  throws  as  ordinates,  it  will 
be  found  to  be  a  straight  line  passing  through  the  origin  ;  thus 
verifying  the  relation 

QB^~  (173) 

KS 

If  the  results  of  (A)  and  (B)  be  combined,  we  have 

Qs=M±f-,  (174) 

Xs 

in  which  the  constant  M  is  defined  as  the  coefficient  of  mutual 
induction  of  the  two  coils.  The  value  of  M  depends  solely 
on  the  construction  of  the  two  coils  and  their  relative  position. 
If  Q,  /,  and  R  be  measured  in  coulombs,  amperes,  and  ohms, 
respectively,  M  will  be  expressed  in  henrys. 

(C)  If  the  distance  between  the  primary  and  secondary  coils 
be  varied,  the  mutual  induction  will  also  vary.     The  relation 
between  these  two  quantities  may  be  experimentally  determined 
as  follows  :    Make  connections  as  above,   place  the  two  coils 
on  a  common  axis,  and  observe  the  throws  of  the  ballistic  galva- 
nometer  needle   corresponding   to   several   different   distances 
between  the  two  coils.     From  (174)  we  have 


(I75) 


If  the  current  which  flows  in  the  primary  while  that  circuit 
is  closed  has  a  constant  value  throughout  the  experiment,  the 
mutual  induction  will  be  proportional  to  the  product  of  resist- 


244  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

ance  in  the  secondary  circuit  and  the  throw  of  the  galvanom- 
eter needle,  and  we  may  write 

MK§RS.  (176) 

These  observations  should  be  repeated  with  a  soft  iron  core 
in  the  primary  coil,  and  curves  platted  with  distances  between 
the  coils  as  abscissas,  and  the  above  product  as  ordinates. 
If  the  coefficient  of  mutual  induction  is  known  for  any  one 
position,  it  can  now  be  computed  for  any  other  position  by 
a  simple  proportion  between  the  known  and  unknown  co- 
efficients, and  the  corresponding  ordinates  to  the  curve. 


CHAPTER    XL 
GROUP  W:    SOUND. 

Measurement  of  pitch  by  the  syren ;  (W2)  Wave  length  by 
Koenigs  apparatus ;  (W3)  Resonance  of  columns  of  air 
with  determination  of  the  velocity  of  sound ;  (W4)  Velocity 
of  sound  in  brass;  (W5)  The  sonometer;  (W6)  Study  of 
the  transverse  vibration  of  cords  and  wires,  Molers  method. 

EXPERIMENT  Wx.     Measurement  of  pitch  by  the  syren. 

This  experiment  consists  in  the  determination,  by  means 
of  a  syren,  of  the  pitch  of  an  organ  pipe,  first  when  closed  at 
one  end  and  then  when  open.  Each  determination  should  be 
made  several  times.  Two  observers  are  needed  to  make  these 
measurements  successfully,  one  devoting  his  attention  to  keep- 
ing the  syren  in  unison  with  the  pipe,  while  the  other  operates 
the  counter  and  observes  the  time.  To  form  an  estimate  of 
the  degree  of  accuracy  that  is  attainable,  several  measurements 
should  be  made  with  a  tuning  fork  of  known  pitch  before  begin- 
ning observations  with  the  pipe.  (See  Glazebrook  and  Shaw, 
p.  222.) 

EXPERIMENT  W2.  Interference  and  measurement  of  wave 
length  by  Koenig's  apparatus. 

In  the  foriTh  of  apparatus  used  a  manometric  capsule  (mlt  m2) 
is  attached  to  one  end  of  each  of  two  tubes.  The  opposite 
ends  of  the  tubes  are  brought  together  at  a  common  opening 
(Fig.  85),  where  some  sounding  body,  such  as  a  tuning  fork  or 
organ  pipe,  is  to  be  placed.  The  two  tubes  are  initially  of  the 

245 


246  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

same  length,  but  one  of  them  (L2)  is  capable  of  adjustment 
so  that  its  length  can  be  increased  by  about  50  cm.  From  each 
of  the  two  capsules  a  tube  (gly  g<^  leads  to  a  small  gas  jet. 
The  latter  will  be  set  in  vibration  when  the  membrane  of  the 
capsule  is  disturbed,  and  can  be  observed  in  a  revolving  mirror- 
There  is  also  a  third  jet  attached  to  g^  which  is  connected,  by 
tubes  of  equal  length,  to  both  capsules  ;  so  that  if  a  pressure  or 
condensation  is  sent  to  it  by  one  of  the  capsules  at  the  same  time 


Fig.   85. 

that  an  equal  rarefaction  is  sent  by  the  other,  the  two  acting 
on  the  flame  at  the  same  instant  will  not  affect  it.  Each  of  the 
single  jets  will,  however,  still  show  the  disturbance.  In  order 
that  a  condensation  may  exist  at  one  capsule  at  the  same  time 
that  a  rarefaction  exists  at  the  other,  it  is  obvious  that  the  two 
tubes  must  differ  in  length  by  one-half  the  wave  length  of  the 
sound  that  is  producing  the  disturbance,  or  by  some  odd  multiple 
of  a  half  wave  length.  This  can  be  brought  about  by  sliding  the 
movable  tube  in  or  out.  When  the  proper  adjustment  is  ob- 
tained, the  jet  that  is  connected  with  both  capsules  should  show 
a  minimum  disturbance.  It  will  not  be  found  possible  to  pro- 
duce complete  quiescence  in  the  image  of  g^  such  as  is  indicated 
in  Fig.  86.  Care  must  be  taken  to  have  the  tubes  which  supply 


SOUND.  247 

this  jet  of  the  same  length,  and  also  to  have  the  pressure  of  gas 
the  same  in  each.  To  adjust  the  pressure,  pinch  shut  one  of 
the  tubes,  and  note  the  height  of 
flame  due  to  the  other ;  then  pinch 
the  second  tube  and  release  the 
first,  and  adjust  the  supply  of  gas 
until  the  height  of  flame  is  the 
same  in  both  cases. 

The  wave  length  being  deter- 
mined as  described  above,  and  the 
pitch  of  the  fork  used  being  known,  Fig-  86. 

the  velocity  of  sound  can  be   computed.    The  result  obtained 
should  be  compared  with  the  velocity  given  by  the  formula 


(177) 

where  t  is  the  temperature  of  the  room.  (See  Anthony  and 
Brackett.)  If  the  sounding  body  used  is  of  unknown  pitch,  the 
wave  length  can  be  determined  as  before,  the  velocity  of  sound 
obtained  from  the  above  formula,  and  from  these  two  quantities 
the  pitch  can  be  computed. 

EXPERIMENT  W3.  Resonance  of  columns  of  air  and  deter- 
mination of  the  velocity  of  sound. 

The  apparatus  for  this  experiment  consists  of  two  or  more 
glass  tubes  containing  water,  and  arranged  so  that  the  level 
of  the  water  can  be  conveniently  altered,  several  tuning  forks  of 
known  pitch,  a  scale  for  measuring  the  distance  from  the  top 
of  the  water  to  the  open  end  of  the  tube,  and  a  thermometer 
for  determining  the  temperature  of  the  air.  A  convenient 
form  of  apparatus  is  that  shown  in  Fig.  87. 

The  experiment  is  to  be  performed  with  each  of  the  tubes 
as  follows  :  Fasten  one  of  the  forks  to  some  firm  support  so 
that  its  prongs  are  immediately  above  the  end  of  the  tube,  and 
set  it  into  vibration  by  a  blow  from  a  wooden  mallet.  Then 
adjust  the  level  of  the  water  until  the  sound  of  the  fork  is  rein- 


248 


JUNIOR   COURSE    IN    GENERAL   PHYSICS. 


forced  by  resonance  from  the  air  column  contained  by  the  tube. 
Several  positions  of  the  water  level  will  probably  be  found 
where  this  reinforcement  takes  place ;  each  should  be  carefully 
located,  and  the  distance  from  the  top  of  the  water  to  the  open 
end  of  the  tube  should  be  measured.  The  distance  between 
two  consecutive  positions  of  the  water  level  which  produce  rein- 


Fig.  87. 


forcement  is  equal  to  one-half  the  wave  length  of  the  tone  given 
by  the  fork.  Since  the  pitch  of  the  fork  is  known,  the  velocity 
of  sound  can  therefore  be  computed.  As  a  check  upon  the 
results  the  velocity  of  sound  may  be  computed  for  the  tempera- 
ture of  the  air  at  the  time  of  the  experiment,  upon  the  assump- 
tion that  the  velocity  at  o°  is  332  m.  per  second.  (See  Anthony 
and  Brackett.) 

The  observations  described  above  should  be  repeated,  in  the 
case  of  at  least  one  of  the  tubes,  with  several  forks  of  different 
pitch. 


SOUND. 


249 


To  investigate  the  influence  of  the  diameter  of  the  tube  upon 
its  behavior  as  a  resonator,  the  following  method  may  be  fol- 
lowed. Use  a  number  of  tubes  varying  widely  in  diameter,  and 
determine  for  each  the  minimum  length  of  air  column  that  will 
reinforce  a  given  fork.  Care  should  be  used  in  this  work  to 
place  the  fork  always  at  the  same  (known)  distance  above  the 
end  of  the  tube.  For  a  tube  of  small  diameter  the  length  of 
air  column  in  this  case  is  a  quarter  wave  length ;  but  as  the 
diameter  is  made  greater,  the  length  will  be  found  to  vary,  this 
variation  being  due  to  the  spreading  out  of  the  sound  waves 
that  travel  up  the  tube  as  they  reach  the  open  end.  Try  to 
determine  the  law  of  this  variation  by  platting  a  curve  with 
diameters  as  abscissas  and  lengths  as  ordinates. 

The  report  should  contain  a  full  explanation  of  the  resonance 
phenomena  observed. 

EXPERIMENT  W4.  Velocity  of  sound  in  brass.  Kundt's 
method. 

A  brass  rod  about  a  meter  long,  Fig.  80,  is  placed  in  a 
horizontal  position,  and  firmly  supported  at  its  center.  To 


Fig.   88. 


one  end  of  the  rod  is  fastened  a  disk  of  cardboard  or  cork, 
whose  diameter  is  almost  equal  to  that  of  a  glass  tube  in 
which  it  is  inserted.  The  opposite  end  of  this  tube  is  closed 
by  means  of  an  adjustable  piston,  so  that  the  length  of  the  air 


250  JUNIOR   COURSE   IN    GENERAL   PHYSICS. 

column  in  the  tube  can  be  altered.  On  setting  the  rod  into 
vibration  (by  rubbing  its  free  end  with  leather  covered  with 
rosin),  the  air  in  the  tube  will  also  vibrate,  and  by  placing  some 
light  powder  in  the  tube  (such  as  lycopodium  or  cork  dust), 
these  vibrations  are  made,  evident  to  the  eye.  If  the  length  of 
the  tube  is  properly  adjusted,  the  dust  will  be  seen  to  distribute 
itself  regularly  in  little  heaps,  these  heaps  corresponding  to 
nodes  in  the  stationary  waves  set  up  in  the  air.  Frequently 
the  dust  figures  are  similar  to  those  shown  in  Fig.  89. 

The  experiment  consists  in  so  adjusting  the  length  of  the  air 
column   as   to   make   this   regular   distribution  of   the  dust  as 


tsn^ 


Fig.  89. 

marked  as  possible.  The  distance  between  consecutive  nodes 
is  then  measured  and  thus  the  wave  length  in  air  of  the  sound 
emitted  by  the  vibrating  rod  is  determined.  Since  the  rod  is 
clamped  at  its  center,  the  wave  length  in  brass  of  the  vibration 
produced  in  it  is  equal  to  twice  the  length  of  the  rod.  (See 
Anthony  and  Brackett.)  Having  the  wave  lengths  of  the  same 
note  in  air  and  in  brass,  the  ratio  of  the  two  gives  the  ratio 
of  the  two  velocities  of  sound. 

To  compute  the  velocity  of  sound  in  air  at  the  temperature 
of  the  experiment,  make  use  of  the  formula  ^  =  332  Vi-f  2T?^ 

EXPERIMENT  W6.     The  sonometer. 

All  the  laws  of  vibrating  strings  or  wires  may  be  expressed 
by  the  formula 


where  n  is  the  number  of  complete  oscillations  per  second,  /  the 
length  of  the  vibrating  segment  of  the  string,  r  its  radius,  d  its 


SOUND. 


25I 


density,  and   T  the  tension  to  which  the  string  is  subjected. 
This  formula  may  also  be  put  in  the  form 


i 

—, 
2  I 


where  m  is  the  mass  of  unit  length.  This  form  of  the  equation 
is  often  the  more  convenient.  (See  Daniell,  Glazebrook,  and 
Shaw,  or  Anthony  and  Brackett.) 

The  object  of  the  experiment  is  to  verify  this  formula  experi- 
mentally.    The  apparatus  used  is  a  sonometer  (Fig.  90),  which 


Fig.  90. 

consists  of  a  long  wooden  box,  upon  which  may  be  stretched  two 
or  more  wires.  One  of  these  wires  is  stretched  by  turning  a  key. 
The  tension  of  the  other  one  must  be  known.  To  secure  this, 
one  end  of  the  string  is  fastened  to  the  box,  and  the  other  end 
to  a  lever  which  moves  about  a  knife-edge  as  an  axis.  From 
the  other  end  of  this  lever  weights  are  suspended.  If  the  two 
lever  arms  are  made  equal,  the  tension  of  the  string  is  equal  to 
the  weight  suspended.  The  length  of  the  vibrating  segment  of 
either  of  these  strings  may  be  varied  by  changing,  the  position 
of  a  movable  bridge. 

In  performing  the  experiment  adjust  the  length  and  tension 
of  the  string  which  is  to  be  studied  until  it  vibrates  in  unison 
with  a  tuning  fork  of  known  pitch.  Having  measured  the 
values  of  r>  /,  71,  etc.,  compute  the  pitch  of  the  string  from 


252 


JUNIOR   COURSE    IN    GENERAL   PHYSICS. 


the  formula  given  above,  and  note  how  closely  the  result  agrees 
with  the  known  pitch  of  the  fork.  The  law  should  be  tested  in 
this  way  for  at  least  three  strings  of  different  diameter  and 
density,  and  several  forks  should  be  used  with  each  string. 

On  account  of  the  great  difference  in  quality  between  the 
note  of  the  string  and  that  of  the  fork,  great  care  must  be  used 
in  adjusting  the  former.  If  the  ear  is  untrained,  a  mistake  of 
an  octave  is  not  unusual. 

EXPERIMENT  W6.  The  study  of  the  transverse  vibration  of 
cords  and  wires,  by  Holer's  method. 

The  apparatus*  for  this  experiment  is  a  modification  of 
Melde's  well-known  device.  It  consists  of  an  instrument  de- 


O 


Fig-  91. 

signed  for  the  purpose  of  maintaining  in  continued  circular 
vibration  a  cord  or  wire,  the  tension  of  which  is  adjusted  by 
weights  until  well-defined  loops  and  nodes  are  produced. 

*  For  a  fuller  description,  see  G.  S.  Moler's  article  in  the  American  Journal  of 
Science,  Vol.  36,  p.  337. 


SOUND. 

The  cord  in  question  is  attached  at  one  end  to  a  crank  of 
small  throw  which  is  driven  at  a  high  speed  by  means  of  an 
electric  motor  or  water  wheel.  The  velocity  is  maintained  con- 
stant by  the  action  of  an  electric  brake. 

The  arrangement  of  these  parts  is  shown  in  Fig.  91.  In 
that  figure,  A  is  the  main  shaft,  and  C  the  crank  pin,  upon  which 
a  hook  is  placed  carrying  the  cord  or  cords  D  to  be  put  into 
vibration. 

To  counteract  the  tension  of  these  cords,  which  would  other- 
wise produce  too  great  friction,  the  crank  is  attached  to  the 
bearing  E  by  a  stout  cord,  the  strain  upon  which  is  adjustable 
by  means  of  a  tightening  pin  at  F. 

In  order  to  drive  simultaneously  two  parallel  cords  at  differ- 
ent speeds,  there  is  a  second  shaft  and  crank  which  can  be  put 
into  motion  at  will,  by  means  of  the  pinion  wheels  at  G  and  H. 


Fig.   92. 


The  cords,  the  vibrations  of  which  are  to  be  studied,  one 
end  of  each  of  which  is  fastened  to  the  crank  hooks,  are  carried 
over  pulleys,  attached  at  any  desired  distance  upon  the  base  of 
the  apparatus  (see  Fig.  92),  and  are  subjected  to  tension  by  the 
application  of  weights.  When  these  weights  are  in  proper  rela- 
tion to  the  velocity  of  the  crank,  the  cord  breaks  into  nodes  and 
falls  into  a  stable  condition  of  vibration  which  is  maintained  as 
long  as  the  conditions  upon  which  it  depends  continue. 

One  of  the  conditions  of  equilibrium  is  the  speed,  and  it  is 
for  the  purpose  of  regulating  this  factor  that  the  electric  brake 
is  used.  This  part  of  the  apparatus  is  shown  in  Figs.  93  and 

94- 

In  the  former  figure,  J  is  a  lever  pivoted  at  K  (see  also  /, 
Fig.  91),  and  this  is  forced  outward  with  increasing  speed  until 


254 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


it  bends  the  spring  L  and  makes  a  contact  with  M.     Thus  an 
electric  circuit  through  the  electromagnet  P  (Fig.  94)  is  closed 

and  the  brake  Q  is  thrown 
against  the  periphery  of  a  wheel 
upon  the  main  shaft.  In  this 
way  the  speed  is  checked  when- 
ever it  exceeds  a  certain  desired 
rate. 

It  should  be  noted  that  in 


Fig.  93. 


practice  this  electric  regulator  has  but  little  to  do,  since  a  heavy 
cord  2  m.  in  length,  or  even  less,  is  in  itself  a  very  effective 
regulator  of  speed  as  soon  as  it  has  once  been  brought  into 
definite  vibration. 

It  is  indeed  entirely  practicable  to  perform  the  experiment 
without  putting  the  brake  into  function,  provided  that  two  cords 
be  used,  one  of  which  is  maintained 
under  constant  tension  and  serves 
as  a  governor  while  variations  of 
length   and  load  are  made  in  the 
case  of  the  other.     The  governing 
cord    should    be    of    considerable 
mass. 

The  experiment  consists  in 
varying  the  factors  upon  which 
the  transverse  vibration  of  strings 
depends,  and  verifying  the  rela- 
tions which  exist  between  them 
and  the  rate. 

These  factors  are : 

(1)  The  cross-section  (Sf),  which 

may  be  conveniently  varied  by  using  several  strands  of  a  light 
cord  in  common. 

(2)  The  length  (L),  which  is  to  be  changed  by  shifting  pul- 
leys along  the  base  of  the  instrument. 

(3)  The  tension. 


Fig.  94. 


SOUND. 


255 


The  rate  itself  may  be  subject  to  one  change  by  fastening 
the  cords  first  to  the  main  shaft  and  then  to  the  shaft  H. 

The  results  are  to  be  arranged  as  shown  in  the  following 
table. 

TABLE. 


Observa- 
tion. 

Cross- 
Section. 
5. 

Length. 

Rate  of 
Vibra- 
tion. 

N. 

Number 
of 
Segments. 
«. 

Tension  in 
Grams. 
/>. 

Square 
Root  of 
Tension. 

Sf. 

Constant 

H      ^^ 

NL     id 

No.      i 

2 

I 

627. 

25.04 

12.52 

"           2 

T3 

2 

2 

162. 

12.70 

12.70 

"       3 

2 

3 

74- 

8.60 

12.90 

"        4 

"to 

2 

4 

41. 

6.40 

1  2.  80 

"       5 

2 

5 

26. 

5.10 

12-75 

No.      6 

T3 
C 

I 

2 

i 

627. 

25.04 

12.52 

"        7 

M          & 

I 

2 

i 

266. 

16.30 

12.22 

"       8 

In 

i 

2 

i 

1  66. 

12.90 

I2.9O 

No.      9 

2 

i 

2500. 

50.00 

12.50 

"        10 

t« 
*1 

2 

2 

625. 

2O.OO 

12.50 

"      ii 

a 

2 

3 

280. 

16.70 

12.52 

"        12 

1/5 

2 

4 

158. 

12.  60 

12.  60 

No.    13 

. 

I 

i 

1  66. 

I2.9O 

12.90 

"      14 

*"O 
c 

M       rt 

I 

2 

41. 

6.40 

1  2.  80 

"      J5 

5 

I 

3 

1  8. 

4.20 

12.  6O 

"      16 

I 

4 

9-5 

3.10 

I2.4O 

If  N  be  the  number  of  vibrations  per  unit  of  time,  L  the 
length  of  the  cord,  n  the  number  of  segments,  and  V  the 
velocity  of  transmission  of  an  impulse  transmitted  to  the  cord, 
we  have  the  familiar  formula  expressing  the  transverse  vibra- 
tions of  flexible  cords  : 


=  JLTV- 

2  L 


(179) 


If  P  is  the  tension  of  the  cord,  s  its  cross-section,  and  d  its 
density,  we  have  also 


sd 


256  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

Finally,  if  X  is  the  wave  length,  we  may  write 

X  =— ,  (181) 

(182) 

-  (183) 


CHAPTER    XII. 


GROUP  X:    LENSES   AND  MIRRORS. 

(X1)  Radius  of  curvature  of  a  lens  (by  reflection) ;  (X2)  Focal 
length  of  a  concave  mirror ;  (X3)  Focal  length  of  a  convex 
lens ;  (X4)  Magnifying  power  of  a  telescope ;  (X6)  Mag- 
nifying power  of  a  microscope  and  focal  lengths  of  same. 

EXPERIMENT  Xr     Determination  of  the  radius  of  curvature 
of  a  lens  by  reflection. 

The  apparatus  consists  of  a  telescope   placed   midway  be- 
tween two  small  gas  jets  (g,  g \  Fig.  95),  the  distance  between 


30 


Fig.  95. 


the  jets  being  capable  of  adjustment.  The  lens  (L,  Figs.  95 
and  96)  whose  curvature  is  desired  is  placed  at  a  distance 
of  from  i  to  2  m.,  and  in  such  a  position  that  the  reflected 
images  of  the  two  flames  can  be  seen  in  the  telescope.  The 
apparent  distance  between  the  images  is  measured  by  means 

257 


VOL.  I  S 


258 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


of  a  scale  (Fig.  97)  fastened  to   the  surface  of  the  lens,  and 
from  this  measurement,  together  with  the  distance  from  the 


Fig.  96- 

lens  to  the  flames,  and  the  actual  distance  between  the  flames, 
the  radius  of  curvature  can  be  computed. 

The  problem  with  which  this  experiment  deals  consists 
in  finding  the  radius  of  curvature  r(=co,  Fig.  98)  in  terms 
of  L,  the  distance  between  the  gas  jets  (gg1) ;  of  Dt  the  dis- 


Fig.  97.  Fig.  98. 

tance  from  telescope  to  lens  (cT),  and  of  s,  the  apparent 
distance  of  the  images  (gn,  gnl)  as  measured  upon  the  scale 
on  the  face  of  the  lens  (aa1). 

From  the  relation  of  conjugate  foci  we  have 

47--L-— i  (184) 

gcl     g"c'         oc 

and  in  case  the  telescope  is  at  a  distance  from  the  lens  much 
greater  than  L,  we  may  write  as  an  approximation 

I    _  I  _'2L 

Tc     tc     oc 
Dr 


or 


where 


(185) 
(186) 


=  ct. 


LENSES   AND   MIRRORS.  259 

From  the  geometry  of  similar  triangles  we  have,  also, 


where  /  =  g",  gn',  which  is  the  distance  between  the  images. 

The  quantities  /  and  d  are  to  be  eliminated,  and  r  is  to  be 
expressed  as  stated  above. 

Combining  equations  187  and  188,  we  have 

s(r+D)     L(r-d]      Lr 


D 


(189) 


'-TIT;  (I90) 

To  obtain  accurate  results,  the  conditions  of  the  experi- 
ment should  be  varied  by  changing  the  position  of  the  lens, 
and  by  altering  the  distance  between  the  flames.  Make  five 
or  six  determinations  for  each  side  of  the  lens  and  use  the 
average  of  each  set. 

It  may  happen  that  two  pairs  of  images  are  seen  by  reflec- 
tion. This  is  due  to  the  fact  that  a  part  of  the  light  from  the 
flames  passes  through  the  first  surface  and  suffers  reflection 
at  the  second.  One  pair  of  images  will  probably  be  erect 
and  the  other  inverted,  so  that  no  difficulty  need  be  experi- 
enced in  distinguishing  between  the  two. 

Addendum  to  the  report : 

Rays  from  the  gas  jet  g  are  reflected  from  the  face  of  the 
lens,  and  enter  the  telescope  T.  The  angles  which  the  incident 
and  reflected  rays  make  with  the  normal  to  the  surface  are 
equal.  From  this  consideration  deduce  formula  190,  without 
using  the  relation  of  conjugate  focal  lengths,  or  the  position  of 
the 


260  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

EXPERIMENT  X2.     Focal  length  of  a  concave  mirror. 

The  object  of  this  experiment  is  to  verify  the  formula  which 
shows  the  relation  between  the  conjugate  foci  and  the  principal 
focus  of  a  concave  mirror ;  viz. 


_     ,  _    =    = 

Pi     Pi     r     f 


(191) 


The  apparatus  required  consists  simply  of  the  mirror  my 
a  glass  scale  S,  a  screen  Sf,  and  a  gas  flame. 

The  mirror  should  first  be  mounted  (see  Fig.  99)  in  such 
a  way  that  its  principal  axis  is  nearly  horizontal.  The  glass 


S'- 


;  Gas  Flame 
Fig.  99- 

scale  *  may  then  be  placed  at  some  point  in  this  axis,  with  the 
gas  flame  a  short  distance  behind  it.  It  will  be  found  more 
convenient  to  work  in  a  room  which  is  partially  darkened. 

The  position  of  the  image  of  the  scale  may  now  be  found  by 
trial,   a  screen   Sf   (preferably  of   ground   glass)   being  placed 

*  A  glass  scale    is    recommended  merely  because  it   constitutes  an  "  object '* 
whose  image  will,  in  general,  be  especially  sharp. 


LENSES   AND   MIRRORS.  26l 

in  such  a  position  that  the  image  thrown  upon  it  is  as  sharp  as 
possible.  This  adjustment  may  be  made  more  accurately  if 
the  mirror  is  partly  covered,  so  that  only  a  comparatively  small 
portion  near  the  center  is  used.  The  distances  of  object  and 
image  from  the  mirror  are  now  to  be  measured,  together  with 
the  distance  between  the  lines  in  the  image  of  the  scale. 
Repeat  these  measurements  for  three  or  four  different  positions 
of  the  scale,  the  position  in  each  case  being  such  that  the 
image  lies  between  the  scale  and  the  lens.  The  focal  length 
and  the  radius  of  curvature  are  to  be  computed  from  each 
of  the  observations. 

As  a  cheek  upon  the  results,  the  center  of  curvature  may 
be  located  by  placing  a  needle,  or  other  pointed  object,  in  such 
a  position  that  the  image  of  its  point  shall  coincide  in  position 
with  the  point  itself.  This  may  be  done  quite  accurately  by 
moving  the  eye  about  and  noting  whether  the  relative  positions 
of  image  and  object  vary. 

Addenda  to  the  report: 

(1)  From  the  data  obtained,  verify  the  formula  which  shows 
the  relation  between  the  size  of  the  image  and  its  distance 
from  the  mirror;    i.e.  if  the  lengths  of  object  and  image  are 
respectively  /x  and  /2, 

l\  :4=/i  :A- 

(2)  Give  a  demonstration  of  the  formula  above  referred  to ; 
also  the  formula  for  conjugate  foci. 

(3)  Indicate  the  advantage  of  using  only  a  small  central 
portion  of  the  surface  of  the  mirror. 

EXPERIMENT  X3.  Determination  of  the  focal  length  of 
a  convex  lens.  ' 

The  focal  length  of  the  lens  used  is  to  be  determined  by 
each  of  the  four  methods  described  below,  a  number  of  obser- 
vations being  made  in  each  case,  and  the  average  used. 


262 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


(i)  The  lens  is  made  to  form  an  image  F,  of  some  object 
whose  distance  is  so  great  that  light  proceeds  from  it  to  the 
lens  in  rays  that  are  very  nearly  parallel.  The  focal  length 
is  then  equal  to  the  distance  between  lens  and  image.  The 
sun  is  usually  the  most  convenient  source  of  light  for  use  in 
this  method,  the  rays  being  rendered  horizontal  by  reflection 
from  a  mirror  M  (Fig.  100).  A  screen  of  ground  glass  or  paper 


Fig.  100. 

is  adjusted  until  the  image  thrown  upon  it  is  as  distinct  and 
sharp  as  can  be  obtained.  The  focal  length  is  then  equal  to 
the  distance  from  the  screen  to  the  center  of  the  lens.  This 
method  is  not  capable  of  great  accuracy,  but  is  more  direct 
than  those  which  follow. 

(2)  A  telescope  which  has  been  focused  for   parallel  rays 
is  used  to  observe  some  sharply  defined  object  as  seen  through 
the  lens.     The  position  of  the  lens  having  been  adjusted  until 
the  object  is  seen  to  be  properly  focused  in  the  telescope,  the 
distance  between  lens  and  object  is  equal  to  the  focal  length 
required.     In   principle   this    method    is   practically   the   same 
as  that  first  described,  and  the  degree  of  accuracy  that  can 
be  attained  is  about  the  same  in  each. 

(3)  An  object  is  placed  at  any  convenient  distance  in  front 
of  the  lens,  and  a  screen  is  adjusted  until  the  image  received 
upon   it   is    sharply  defined.     The   focal   length    can    then    be 
computed  from   measurements  of  the  distances   of  object  and 


LENSES   AND   MIRRORS. 


263 


image  from  the  lens.  If  /x  and  /2  are  these  two  distances, 
we  have 

7^7  •      (I92) 

The  luminous  object  used  may  be  the  flame  of  a  candle  or 
gas  jet.  There  are  some  objections,  however,  to  the  use  of  a 
flame,  on  account  of  the  flickering  caused  by  air  currents. 
Better  results  can  usually  be  obtained  by  using  a  fine  thread 
or  wire  which  is  stretched  across  an  opening  in  an  opaque 
screen  (Fig.  101).  When  the  aperture  is  illuminated  by  means 


Fig.   101. 

of  a  lamp,  the  shadow  of  the  wire  forms  an  image  which 
is  unaffected  by  the  flickering  of  the  flame,  and  which  can  be 
very  sharply  focused. 

(4)  Placing  the  object  at  any  convenient  distance  from  the 
lens,  adjust  the  position  of  the  screen  until  the  image  is  sharply 
focused.  Then,  without  changing  the  position  of  the  screen, 
move  the  lens  until  a  second  position  is  found,  such  that  a 


264  JUNIOR   COURSE    IN   GENERAL   PHYSICS. 

sharp  image  is  formed.  From  the  distance  between  object 
and  screen,  and  the  distance  through  which  the  lens  is  moved, 
the  focal  length  can  be  computed.  If  /  and  a  are  the  two 
distances, 


This  method  of  determining  focal  length  has  the  advantage 
of  being  uninfluenced  by  any  uncertainty  as  to  the  thickness  of 
the  lens  and  the  position  of  the  principal  points.  Since  it  is 
merely  the  distance  through  which  the  lens  is  mcved  that 
is  required,  measurements  can  be  made  to  any  convenient 
point  on  the  support  of  the  lens,  and  no  correction  need  be 
made  for  the  thickness  of  the  glass.  For  this  reason  the 
method  will  probably  give  better  results  than  can  be  obtained 
by  any  of  the  three  methods  first  described. 

Addenda  to  the  report  : 

(1)  Sharper  images,   and  therefore  more   accurate   results, 
will  be  obtained  if  the  lens  is  covered,  so  that  only  a  small 
region  near  the  center  is  used.     (Explain.) 

(2)  From  the  curvature  of  each  face  of  the  lens  and  your 
determination  of  the  focal  length,  compute  the  index  of  refrac- 
tion of  the  glass  from  which  it  is  made. 

'EXPERIMENT  X4.     Magnifying  power  of  a  telescope. 

Focus  the  telescope  upon  some  large  object,  such  as  a  scale, 
which  contains  sharply  defined  portions  of  equal  length.  The 
bricks  in  the  wall  of  a  building,  or  the  pickets  of  a  fence,  will 
serve  for  this  purpose.  Looking  through  the  telescope  with 
both  eyes  open,  the  magnified  image  of  the  scale  will  be  seen 
by  one  eye,  while  with  the  other  the  scale  is  observed  directly. 
By  a  comparison  of  the  two  images  the  magnifying  power  is 
determined.  For  example,  if  one  division  of  the  image  seen 
in  the  telescope  covers  ten  divisions  of  the  unmagnified  image, 


LENSES   AND   MIRRORS. 


265 


the  magnifying  power  is  ten.  To  guard  against  errors  due  to 
a  difference  in  the  two  eyes,  it  is  best  to  use  the  left  eye  in 
observing  the  telescopic  image  as  often  as  the  right. 

The  magnifying  power  should  be  determined  in  this  way 
when  the  object  observed  is  at  several  different  distances, 
ranging  from  a  distance  that  is  so  great  as  to  be  practically 
infinite,  to  the  least  distance  for  which  the  telescope  can  be 
focused.  If  any  difference  is  found  in  the  magnifying  power, 
the  variation  should  be  shown  by  a  curve  in  which  distances 
and  magnifying  powers  are  used  as  co-ordinates. 

For  some  one  distance  of  the  object  observed  the  distances 
between  the  various  lenses  should  be  accurately  measured  when 

the  telescope  is  focused. 

• 

Addenda  to  the  report : 

(1)  Determine  the  focal  length  of  each  of  the  lenses,  and 
compute  the  magnifying  power,  drawing  a  diagram  to  scale  to 
show  the  position  and  size  of  the  various  images. 

(2)  Explain  the  cause  of  the  variation  in  magnifying  power 
with  the  distance  of  the  object. 

EXPERIMENT  X5.  Magnifying  power  of  a  microscope  and 
determination  of  the  focal  length  of  its  lenses. 

I. 

The  "open-eye"  method. 

This  method  is  similar  to  that  described  for  the  determin- 
ation of  the  magnifying  power  of  a  telescope. 

(i)  Focus  the  microscope  upon  a  finely  divided  scale  and 
place  another  scale  at  the  side  of  the  instrument  at  a  distance 
from  the  eye  equal  to  the  distance  of  distinct  vision  (about 
25  cm.).  By  observing  the  scale  with  one  eye  and  the  image 
formed  in  the  microscope  with  the  other,  the  apparent  size 
of  the  magnified  image  is  determined.  The  ratio  of  this 
to  the  actual  size  is  the  magnifying  power. 


266  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

(2)  Measure   the   distance   between   the    object   glass    and 
eye  piece  and  determine  the  focal  length  of  the  latter  by  one 
of  the  methods  of  Exp.  X3.     From  a  knowledge  of  the  magni- 
fying  power   it   will    now   be   possible   to    compute   the   focal 
length  of  the  object  glass. 

(3)  Construct  a  diagram  to  scale  to  explain  the  action  of 
the  instrument,  showing  the  position  and  size  of  each  image. 
(See  Gage,  Microscopical  Methods,  p.  65.) 

II. 

Franklin  s  method. 

The  object  of  this  method  is  to  find  the  focal  lengths  of 
microscope  lenses  from  the  magnifying  power  with  short  and 
long  draw  tubes. 

The  apparatus  required  is  a  compound  microscope,  a  microm- 
eter caliper,  a  stage  micrometer,  an  unsilvered  glass  microscope 
slip,  and  a  scale  divided  to  millimeters. 

The  experiment  consists  in  the  following  determinations  : 

(1)  The  magnifying  power  (m)  of  the  microscope  with  short 
tube. 

(2)  The  magnifying  power  (m1)  with  long  tube. 

(3)  The  change  (/)  in  the  length  of  the  microscope  tube. 

(4)  The  movement  of  the  objective  in  refocusing  for  deter- 
mination (2).    This  measurement  is  to  be  made  by  means  of  the 
caliper. 

The  magnifying  power,  as  in  the  previous  method,  is  the 
ratio  of  the  apparent  size  of  an  object  as  seen  with  the  micro- 
scope to  its  apparent  size  as  seen  with  the  naked  eye  at  a 
distance  of  D  centimeters. 

Instead  of  measuring  magnifying  power  by  the  "  open-eye  " 
method,  the  glass  slip  is  mounted  obliquely  before  the  eye  piece 
(Fig.  102),  so  as  to  bring  an  image  of  the  scale  (S)  into  the  field. 

The  theory  of  the  method  of  computing  focal  lengths  is  as 
follows. 

Let  /  be  the  equivalent  focal  length  of  the  eye  piece  ;  /',  the 


LENSES    AND   MIRRORS. 


267 


focal  length  of  the  objective;  a,  the  distance  from  the  object  to 
the  center  of  the  objective ;  and  b,  the  distance  of  the  image 
from  the  center  of  the  objective. 


I 


s' 


Fig.    102 

We  have  the  following  equations  : 


m 


=(!+')-> 

\p        Ja 

'-(-+*)—> I- 

\p        Ja-k 


£ 

V 


p     a-k     b+f 

from  which  these  may  be  obtained. 


2km' 
m'  —  m 


SV  __ 


(m'  — 


=  0. 


(194) 

(195) 
(196) 

(197) 

(198) 
(199) 


From  equation   199,  a  may  be  computed,  then  b  from  (198), 
then  finally  /  and  p'  by  combining  (194)  or  (195)  with  (196)  or 

(197). 


268  JUNIOR   COURSE   IN  GENERAL   PHYSICS. 

The  distance  D  is  the  distance  from  the  focal  plane  in  the 
eye  piece  (which  in  the  case  of  a  negative  [Huyghenian]  eye 
piece  lies  midway  between  the  two  lenses,  and  in  the  positive 
eye  piece  is  distant  from  the  eye  lens  by  an  amount  equal  to 
three-fourths  of  the  distance  between  the  lenses)  to  the  scale 
(s),  Fig.  102. 

GROUP  Y :  THE  SPECTROSCOPE  AND  PHOTOMETER. 

(Yj)  Index  of  refraction  of  a  prism  ;  (Y2)  Flame  spectra  of  the 
metals;  (Y3)  Distance  between  the  lines  of  a  grating; 
(Y4)  The  Bunsen  photometer. 

EXPERIMENT  Yr  Measurement  of  the  index  of  refraction 
of  a  prism  by  means  of  a  spectrometer. 

In  the  spectrometer  used  (see  Fig.  103),  the  rays  of  light 
from  the  source  are  made  to  pass  through  a  narrow  vertical  slit, 


Fig.    103. 

and  are  then  rendered  parallel  by  means  of  a  lens.  This  lens 
and  slit,  mounted  in  a  tube  to  exclude  stray  light,  constitute 
what  is  known  as  the  collimator.  After  leaving  the  collimator, 
the  light  is  made  to  pass  through  a  prism,  and  is  finally  observed 
by  means  of  a  telescope.  Both  collimator  and  telescope  are 
adjusted  so  as  to  point  towards  the  center  of  a  horizontal  grad- 
uated circle,  and  the  telescope  is  free  to  rotate  about  a  vertical 


THE   SPECTROSCOPE   AND   PHOTOMETER. 


269 


axis  passing  through  this  center.  By  means  of  a  vernier 
attached  to  its  support,  the  angular  position  of  the  telescope 
can  be  read. 

Before  beginning  observations  with  the  prism,  the  collimator 
must  be  so  adjusted  that  the  light  leaves  it  in  parallel  rays. 
To  accomplish  this,  take  the  instrument  to  an  open  window  and 
focus  the  telescope  on  some  object  which  is  so  far  away  that 
the  rays  from  it  are  practically  parallel.  Then  turn  the  tele- 
scope so  that  it  points  directly  at  the  collimator,  and,  without 
changing  the  focus  of  the  telescope,  alter  the  length  of  the 
collimator  tube  until  the  image  of  the  slit,  as  seen  in  the  tele- 


Fig.  104. 

scope,  is  sharply  denned.  Both  telescope  and  collimator  are 
now  properly  focused,  and  should  not  be  altered  during  the 
experiment. 

In  order  to  obtain  the  index  of  reflection,  it  is  necessary 
to  know  the  angle  of  the  prism,  and  the  angle  through  which 
it  bends  the  %'ays  from  the  collimator  when  in  the  position 
of  "  minimum  deviation." 

(i)  To  determine  the  angle  of  the  prism.  —  Place  the  prism 
near  the  center  of  the  graduated  circle,  with  its  refracting  edge 
turned  toward  the  collimator  (Fig.  104).  Turn  the  telescope 


2/0  JUNIOR  COURSE    IN   GENERAL   PHYSICS. 

until  the  slit  is  seen  by  reflection  from  one  face  of  the  prism, 
and  adjust  the  position  of  the  telescope  until  its  cross-hair 
coincides  with  the  image  of  the  slit.  Record  the  position  of 
the  telescope  as  read  by  the  vernier.  Then  set  the  cross-hairs 
in  the  same  way  upon  the  image  of  the  slit,  as  reflected  from 
the  other  face  of  the  prism,  position  Tf,  and  again  read  the 
vernier.  The  angle  between  the  two  positions  of  the  telescope 
is  then  equal  to  twice  the  angle  of  the  prism.  A  number  of 
readings  should  be  taken. 

(2)  To  determine  the  angle  of  minimum  deviation.  —  Up  to 
this  point  in  the  experiment  any  source  of  light  will  serve 
equally  well ;  but  since  different  colors  are  bent  by  refraction 
in  different  degrees,  it  will  now  be  necessary  to  use  some  mono- 
chromatic light.  The  most  convenient  light  of  a  single  color 
is  that  obtained  by  burning  some  salt  of  sodium  in  the  Bunsen 
flame.* 

Place  the  prism  near  the  center  of  the  horizontal  circle, 
and  in  such  a  position  that  light  from  the  collimator  will  be 
refracted  through  it  and  pass  into  the  telescope.  If,  while 
observing  the  image  of  the  slit  in  the  telescope,  the  table  which 
carries  the  prism  is  slowly  turned,  the  image  will  be  seen  to 
move  across  the  field,  and  it  may  be  necessary  to  shift  the 
position  of  the  telescope  in  order  to  keep  it  in  sight.  In  this 
way  the  prism  can  be  set  by  trial  to  the  position  which  causes 
the  light  to  deviate  least  from  its  original  direction  on  leaving 
the  collimator ;  when  this  position  is  reached,  a  slight  motion 
of  the  table  in  either  direction  will  cause  the  image  to  move 
toward  a  position  of  greater  deviation.  When  this  setting  is 
made  as  accurately  as  possible,  bring  the  cross-hair  into  coinci- 
dence with  the  image  of  the  slit,  and  take  the  reading  of  the 
vernier.  Several  settings  should  be  made  in  this  way,  with 

*  If  light  of  a  different  wave  length  were  used,  the  index  of  refraction  obtained 
would,  of  course,  be  different.  Since  this  experiment  is,  however,  merely  intended 
to  illustrate  the  use  of  the  spectrometer,  it  will  be  found  best  to  use  the  most  con- 
venient monochromatic  light;  viz.  sodium. 


THE    SPECTROSCOPE   AND   PHOTOMETER.  271 

deviations  both  to  the  right  and  left.     The  index  of  refraction 
can  be  then  computed  from  the  formula 


> 
sm  |-  a 

in  which  a  is  the  angle  of  the  prism,  and  8  the  angle  of  mini- 
mum deviation. 

Indices  of  refraction  are  to  be  obtained  in  this  way  for 
several  prisms  of  different  materials. 

EXPERIMENT  Y2.  Study  of  the  flame  spectra  of  various 
metals. 

This  experiment  should  follow  that  on  the  determination  of 
indices  of  refraction  (No.  Yj),  and  can  be  more  conveniently 
performed  if  there  are  two  observers.  The  ^apparatus  required 
is  a  spectrometer,  or  spectroscope,  similar  to  the  one  used  in 
that  experiment. 

The  substances  whose  flame  spectra  are  most  readily  studied 
are  Na,  Li,  K,  Sr,  Ba,  Ca,  Rb,  and  Cs.  A  salt  of  one  of  these 
metals,  usually  either  the  chloride  or  the  carbonate,  is  placed  in 
the  colorless  flame  of  a  Bunsen  burner,  immediately  in  front  of 
the  slit  of  the  spectrometer.  The  spectrum  of  the  flame,  when 
colored  by  the  incandescent  vapor  of  the  metal,  will  be  seen  to 
consist  of  several  bright  lines,  whose  color  and  arrangement 
are  characteristic  of  the  metal  studied.  To  map  the  position  of 
the  different  lines,  the  prism  should  first  be  adjusted  to  the 
position  of  minimum  deviation  for  sodium  light.  Then,  without 
altering  the  adjustment  of  the  prism,  set  the  telescope  successively 
on  each  of  the  lines  of  the  spectrum,  reading  its  angular  position 
by  means  of  the  vernier.  The  spectrum  can  be  mapped  for 
comparative  purposes  from  these  results.  The  approximate 
limits  of  the  visible  spectrum,  as  determined  by  substituting 
white  light  for  the  metallic  spectrum,  should  be  marked  on  each 
diagram,  and  the  colors  and  relative  intensities  of  the  lines 
should  be  indicated.  If  it  is  desired  to  determine  the  wave 


272  JUNIOR   COURSE   IN    GENERAL   PHYSICS. 

length  of  each  of  the  lines,  a  grating  may  be  used  instead  of  a 
prism;  or,  if  this  is  not  convenient,  the  prism  may  be  "cali- 
brated "  by  reference  to  the  Fraunhofer  lines.  To  accomplish 
this,  the  slit  should  be  illuminated  by  bright  daylight,  or  direct 
sunlight,  and  adjusted  until  the  dark  lines  in  the  spectrum  are 
clearly  seen.  The  prism  is  then  set  to  the  position  of  least 
deviation  for  the  D  line,  and  the  angular  position  of  the  tele- 
scope is  observed  for  several  of  the  more  prominent  lines.  The 
wave  lengths  corresponding  to  these  lines  being  known,  a  curve 
can  now  be  constructed,  in  which  angles  of  deviation  are  taken 
as  abscissas,  and  wave  lengths  as  ordinates.  By  reference  to 
this  curve,  the  wave  length  corresponding  to  any  observed 
deviation  is  readily  determined. 

The  most  instructive  method  of  mapping  bright  line  spectra 
is  that  employed  by  Lecoq  de  Boisbaudran  in  his  work  on  the 


Fig.  105. 

spectra  of  the  metals.*  An  example  is  given  in  Fig.  105.  As 
will  be  seen  from  the  diagram,  each  spectrum  is  mapped  twice, 
once  above  and  once  below  the  median  line.  The  former  xgives 
the  normal,  the  latter  the  prismatic  spectrum  of  the  substance 
in  question.  This  method  should  be  employed  in  reporting 
upon  the  results  of  this  experiment. 

Considerable  difficulty  is  sometimes  met  with  in  working 
with  flame  spectra  in  obtaining  sufficient  permanence  and 
brilliancy  for  accurate  observation.  To  obtain  the  best  results 
the  methods  of  heating  must  be  suited  to  the  salt  used.  In 
some  cases,  a  small  amount  of  the  salt,  when  placed  on  a  wire 

*  Spectres  Lumineux;  Lecoq  de  Boisbaudran,  Paris,  1874. 


THE    SPECTROSCOPE  AND   PHOTOMETER.  273 

and  heated  in  the  flame,  will  form  a  bead  which  lasts  for  a 
considerable  time  and  gives  a  good  spectrum.  In  other  cases 
the  supply  of  salt  will  need  to  be  constantly  renewed.  A  piece 
of  asbestos,  or  wire,  which  has  been  moistened  by  a  strong 
solution  of  the  salt,  will  sometimes  give  good  results.  In 
general,  the  results  will  be  more  satisfactory  when  the  flame 
is  quite  hot.  For  this  reason,  the  substitution  of  a  blast  lamp 
for  the  ordinary  Bunsen  burner  is  sometimes  advisable.  The 
observations  can  usually  be  made  more  rapidly  if  one  observer 
devotes  his  attention  to  keeping  the  flame  in  proper  condition, 
while  the  other  observes  the  spectrum. 

EXPERIMENT  Y3.  Determination  of  the  distance  between  the 
lines  of  a  grating  by  the  diffraction  of  sodium  light. 

The  object  of  this  experiment  is  to  illustrate  the  principles 
involved  in  the  formation  of  spectra  by  a  diffraction  grating. 
It  is  expected,  therefore,  that  the  report  should  contain  a  clear 
explanation  of  the  phenomena  observed.  (See  Kohlrausch, 
Glazebrook's  Physical  Optics,  p.  183,  and  other  books  on  light.) 

The  apparatus  consists  of  a  horizontal  arm,  which  may  for 
convenience  be  provided  with  a  scale,  mounted  upon  a  suitable 
support,  and  having  at  its  center  a  narrow  vertical  slit  which 
may  be  illuminated  by  sodium  light.  To  obtain  the  pure 
yellow  light  of  sodium  it  is  only  necessary  to  place  a  wire 
carrying  a  bead  of  some  sodium  salt  in  the  flame  of  a  Bunsen 
burner.  The  grating  to  be  studied  is  placed  in  front  of  the 
slit  with  its  rulings  vertical,  and  should  be  mounted  on  some 
support  so  that  its  distance  from  the  slit  can  be  varied. 

On  looking  through  the  grating,  several  images  of  the  slit 
will  be  seen  on  either  side,  the  distance  between  these  images 
depending  upon  the  distance  apart  of  the  lines  of  the  grating. 
If  white  light  were  used  instead  of  the  sodium  flame,  these 
images  would  become  spectra.  By  measuring  the  distance 
between  the  grating  and  the  slit  (D,  Fig.  106),  and  the  displace- 
ment of  one  of  the  images  d,  the  angle  through  which  the  ligh 

VOL.   I  —  T 


~*         A 


274 


JUNIOR   COURSE   IN   GENERAL   PHYSICS. 


is  bent   by  diffraction   can   be  determined.     From   this   angle, 
and  the  wave  length  of  sodium  light,  the  distance  between  the 

lines  of  the  grating  is  to  be 
computed. 

In  measuring  the  displace- 
ment of  the  images  the  fol- 
lowing method  will  be  found 
convenient :  Adjust  a  small 
rider,  which  can  be  clamped 
to  the  horizontal  arm,  until  it 
coincides  with  the  correspond- 
ing image  on  the  opposite  side 
of  the  slit.  The  distance  be- 
tween the  two  riders  is  then 
equal  to  twice  the  displacement  of  the  image.  Measurements 
should  be  made  in  this  way  for  four  or  five  different  positions 
of  the  grating  and  with  different  pairs  of  images.  If  the  meas- 
urements are  carefully  made,  the  individual  results  should  agree 
fairly  well,  and  the  final  average  will  not  be  far  from  the  truth. 
The  wave  length  of  sodium  light  may  be  taken  as  0.000059  cm. 


G 


Fig.  106. 


EXPERIMENT   Y4.      Measurement  of   candle   power   by  the 
Bunsen  photometer. 

In  the   Bunsen  photometer   a   screen    of   white   paper  (D, 
Fig.  107),  a  portion  of  which  has  been  made  translucent  by  the 


LX- *-- 


Fig.    107. 


application  of   oil,    is    placed  in   a  blackened    box    (technically 
called  the  carriage),  and  mirrors,  M,M',  are  adjusted   so  that 


THE    SPECTROSCOPE   AND   PHOTOMETER. 


2/5 


both  sides  of  the  paper  may  be  observed  at  the  same  time. 
By  means  of  openings  in  the  carriage,  light  is  admitted  from  the 
two  sources  whose  intensities  are  to  be  compared.  The  carriage 
being  placed  between  the  two  lights,  each  face  of  the  screen 
is  illuminated  only  by  light  from  the  source  toward  which  it 
is  turned,  while  the  translucent  portion  of  the  paper  receives 
light  from  both  sources.  In  using  the  instrument,  the  carriage 
is  shifted  in  position  until  both  sides  of  the  screen  are  seen  to 
be  equally  illuminated.  The  distances  of  the  two  lights  from 
the  screen  are  then  measured,  and  the  relative  intensities  of  the 
two  sources  are  computed  by  the  law  of  inverse  squares. 


Transmitted  Light  Reflected; Light 

Fig.    108. 

The  translucent  spot  on  the  screen  merely  serves  to  locate 
the  position  of  equal  illumination  with  greater  accuracy  than 
could  otherwise  be  obtained.  If  the  adjustment  is  not  quite 
correct,  this  spot  will  appear  dark  on  one  side  and  bright  on 
the  other  (see  Fig.  108)  ;  but  when  the  proper  position  has 
been  found,  it  will  almost  entirely  disappear. 

The  standard  source  of  light  in  the  apparatus  used  consists 
of  an  Argand  burner  placed  just  back  of  an  opaque  screen,  in 
which  a  small  oblong  slit  has  been  cut.  The  slit  is  made  so 
small  that  it  appears  entirely  covered  with  light  when  viewed 
from  the  front,  and  should  be  so  adjusted  as  to  receive  only 
that  light  which  comes  from  the  central  portion  of  the  flame. 
In  this  way  the  irregularities  due  to  the  flickering  of  the  edges 
of  the  flame  are  avoided.  It  is  to  be  observed  that  under  the 
conditions  mentioned,  the  slit  itself  is  the  source  of  light. 


276  JUNIOR   COURSE   IN   GENERAL   PHYSICS. 

Distances  should  be  measured  to  the  plane  of  the  slit,  and  not 
to  the  center  of  the  flame. 

To  perform  the  experiment,  place  the  standard  and  the 
light  to  be  measured  at  opposite  ends  of  the  photometer  bar, 
and  before  beginning  any  actual  observations,  practice  setting 
the  carriage  to  the  position  of  equal  illumination  until  nearly 
the  same  reading  is  obtained  several  times  in  succession. 
After  each  reading,  the  carriage  should  be  shifted  two  or 
three  feet,  in  some  cases  to  the  right  and  in  others  to 
the  left  of  the  proper  setting,  and  then  brought  back  again, 
without  reference  to  the  previous  reading,  until  the  two 
sides  of  the  screen  appear  to  be  equally  illuminated.  The 
uncertainties  of  the  observation,  together  with  slight  variations 
in  the  intensities  of  the  two  lights,  will  make  it  impossible  to 
obtain  coincident  settings,  but  after  a  little  practice  the  succes- 
sive readings  should  agree  to  within  three  or  four  per  cent. 
Constant  differences  are  often  observed  between  the  settings 
of  different  persons.  These  are  due  to  differences  in  the  eye, 
and  cannot  be  avoided. 

After  sufficient  practice  has  been  gained  in  reading,  the 
photometer  may  be  used  to  measure  candle  power  in  some 
one  of  the  cases  which  follow.  It  is  to  be  observed  that  the 
scale  on  the  bar  is  located  without  regard  to  the  positions 
of  the  two  lights,  so  that  suitable  corrections  will  have  to 
be  made  at  each  end. 

(1)  By  comparison    with   a   standard    candle   the   absolute 
intensity  of  the  standard  light  may  be  determined.     At  least 
ten  or  twelve  readings  should  be  taken  to  get  a  good  result. 

(2)  The  light  from  a  fish-tail  burner  or  an  oil  lamp  may  be 
measured  as  the  flame  is  seen  from  different  directions.     It  will 
probably  be  found  that  the  flame  differs  in  illuminating  power 
according  as  the  broad  surface  or  the  edge  is  turned  toward 
the   photometer.     To    investigate   this  variation    first    set   the 
flame  so  that  its  plane  is  parallel  with  the  photometer  bar,  and 
measure    its    intensity.       Then    turn    it    about    a   vertical    axis 


THE    SPECTROSCOPE   AND    PHOTOMETER.  277 

and  measure  the  candle  power  at  intervals  of  30°  until  the 
flame  has  been  turned  through  a  complete  revolution.  The 
results  should  be  shown  graphically  by  a  curve,  in  which 
the  angular  position  of  the  flame,  and  the  observed  intensity 
of  the  light,  are  used  as  polar  co-ordinates.  Such  a  curve  is 
very  commonly  used  to  show  the  distribution  of  the  light  from 
any  source,  and  has  the  advantage  of  showing  at  a  glance  the 
intensity  of  the  light  in  all  horizontal  directions. 

(3)  The  absorbing  power  of  substances  which  are  nearly 
transparent  may  be  determined.  To  accomplish  this,  measure 
the  intensity  of  any  source  as  seen  direct ;  then  interpose  the 
substance  to  be  investigated,  and  see  how  much  the  light  is 
diminished.  From  the  two  measurements  the  percentage 
absorption  can  be  computed.  Investigate  in  this  way  the 
absorption  of  sheets  of  glass  of  different  thickness,  and  of 
cells  containing  various  liquids.  It  must  be  remembered, 
however,  that  some  of  the  light  which  is  apparently  absorbed 
is  really  lost  by  reflection.  If  it  is  desired  to  separate  the 
effects  of  reflection  and  absorption,  more  elaborate  methods 
will  be  necessary. 

Numerous  other  interesting  problems  will  suggest  them- 
selves in  the  solution  of  which  the  photometer  may  be  used. 
For  further  details  concerning  photometry,  see  Vol.  II  of  this 
Manual ;  also  Palaz,  Photometric  Industrielle. 


TABLES. 


[In  these  tables  the  admirable  arrangement  made  use  of 
in  Bottomley's  Four-Figure  Mathematical  Tables  has  been 
followed.] 


280 


LOGARITHMS. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

123 

456 

789 

10 

oooo 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

4  8  12 

17  21  25 

29  33  37 

11 

12 
13 

0414 

0792 

"39 

0453 
0828 

"73 

0492 
0864 
1206 

Q531 
0899 
1239 

0569 

0934 
1271 

0607 
0969 
I3°3 

0645 
1004 
1335 

0682 
1038 
1367 

0719 
1072 
1399 

0755 
1  106 

1430 

4811 

3  7  I0 
3  6  10 

15  19  23 

14  17  21 

13  16  19 

26  30  34 
24  28  31 
23  26  29 

14 
15 
16 

1461 
1761 
2041 

1492 
1790 
2068 

1523 
2095 

1553 
1847 

2122 

1584 
2148 

1614 
1903 

2175 

1644 
i93i 

2201 

1673 
1959 
2227 

i7°3 
1987 

2253 

1732 
2014 
2279 

369 
368 

3  5  8 

12  15  18 

II  14  17 

ii  13  16 

21  24  27 

20  22  25 

18  21  24 

17 
18 
19 

2304 

2553 
2788 

2330 

2577 
2810 

2355 
2601 

2833 

2380 
2625 
2856 

2405 
2648 
2878 

2430 
2672 
2900 

2455 
2695 

2923 

2480 
2718 
2945 

2504 
2742 
2967 

2529 
2765 
2989 

2  5  7 
257 
2  4  7 

IO  12  15 

9  12  14 
9  "  13 

17  20  22 

16  19  21 
16  18  20 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

246 

8  ii  13 

15  17  19 

21 
22 
23 

3222 
3424 
3617 

3243 
3444 
3636 

3263 
3464 
3655 

3284 
3483 
3674 

33°4 
3502 
3692 

3324 
3522 

3711 

3345 
354i 
3729 

3365 
356o 
3747 

3385 
3579 
3766 

3404 
3598 
3784 

246 
246 
246 

8  10  12 
8  10  12 

7  9  ii 

14  16  18 
H  15  17 
13  15  17 

24 
25 
26 

3802 
3979 
415° 

3820 

3997 
4166 

3838 
4014 
4183 

3856 
4031 
42OO 

3874 
4048 
4216 

3892 
4065 
4232 

3909 
4082 

4249 

3927 
4099 
4265 

3945 
4116 
4281 

3962 

4133 
4298 

2  4  5 
2  3  5 
235 

7  9  " 
7  9  10 
7  8  10 

12  14  16 

12  14  15 
II  13  15 

27 
28 
29 

4314 
4472 
4624 

4330 
4487 

4639 

4346 
4502 

4654 

4362 

4518 
4669 

4378 
4533 
4683 

4393 
4548 
4698 

4409 
4564 
4713 

4425 
4579 
4728 

4440 
4594 
4742 

445  6 
4609 

4757 

2  3  5 
2  3  5 
i  3  4 

689 
689 
679 

II  13  14 
II  12  14 
IO  12  13 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

i  3  4 

679 

10  ii  13 

31 
32 
33 

4914 
5Q51 

5i85 

4928 

5065 
5198 

4942 
5079 
5211 

4955 
5092 
5224 

4969 
5105 
5237 

4983 
5ir9 
5250 

4997 
5132 
5263 

5011 

5H5 
5276 

5024 

5159 
5289 

5038 
5172 
5302 

i  3  4 
i  3  4 
1  3  4 

678 
5  7  8 
568 

IO  II  12 
9  II  12 

9  10  12 

34 
35 
36 

5315 
544i 
5563 

5328 
5453 
5575 

5340 
5465 
5587 

5353 
5478 
5599 

5366 

5490 
5611 

5378 
5502 
5623 

539i 
55J4 
5635 

5403 

5527 
5647 

5416  5428 

5539  555i 
5658]  5670 

i  3  4 

I  2  4 
I  2  4 

568 
5  6  7 
5  6  7 

9  10  ii 
9  10  ii 
8  10  ii 

37 
38 
39 

5682 
5798 
59" 

5694 
5809 
5922 

5705 
5821 

5933 

5717 
5832 
5944 

5729 
5843 
5955 

5740 

5855 
5966 

5752 
5866 

5977 

5763 
5877 
5988 

5999 

5786 

5899 
6010 

I  2  3 
I  2  3 
I  2  3 

5  6  7 
5  6  7 
457 

8  9  10 
8  9  10 
8  9  10 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

I  2  3 

4  5  6 

8  9  10 

41 
42 
43 

6128 
6232 
6335 

6138 
6243 

6345 

6149 
6253 
6355 

6160 
6263 
6365 

6170 
6274 
6375 

6180 
6284 
6385 

6191 

6294 
6395 

6201 
6304 
6405 

6212 
63^ 
6415 

6222 
6325 
6425 

I  2  3 
I  2  3 
I  2  3 

4  5  6 
4  5  6 
4  5  6 

7  8  9 
789 
7  8  9 

44 
45 
46 

6435 
6532 
6628 

6444 
6542 
6637 

6454 
6& 

6464 
6561 
6656 

6474 

657i 
6665 

6484 
6580 
6675 

6493 
6590 
6684 

6503 

6599 
6693 

65:3 
6609 
6702 

6522 
6618 
6712 

I  2  3 
I  2  3 
I  2  3 

4  5  6 
4  5  6 
4  5  6 

789 
7  8  9 
7  7  8 

47 
48 
49 

6721 
6812 
6902 

6730 
6821 
6911 

6739 
6830 
6920 

6749 
6839 
6928 

6758 
6848 

6937 

6767 
6857 
6946 

6776 
6866 
6955 

6785 
6875 
6964 

6972 

6803 
6893 
6981 

I  2  3 
I  2  3 
I  2  3 

4  5  5 
4  4  5 
445 

678 
678 
678 

50 

6990 

6998 

7007 

7016 

7024 

7°33 

7042 

7050 

7059 

7067 

I  2  3 

3  4  5 

678 

51 
52 
53 

7076 
7160 
7243 

7084 
7168 
7251 

7093 

7177 

7259 

7101 

7185 
7267 

7110 
7193 
7275 

7118 
7202 
7284 

7126 
7210 
7292 

7135 
7218 
7300 

7J43 

7226 
73o8 

7152 
7235 
73^6 

I  2  3 
I  2  2 
122 

3  4  5 
3  4  5 
3  4  5 

678 
6  7  7 
667 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

738o 

7388 

7396 

1223  4  5 

667 

LOGARITHMS. 


281 


0 

1 

2 

3 

4 

5 

6 

7 

7459 

1  8 

9 

123 

456 

789 

55 

7404 

7412 

7419 

7427 

7435 

7443 

745  i 

7466 

7543 
7619 
7694 

7474 

122 

3  4  5 

5  6  7 

5  6  7 
5  6  7 
5  6  7 

56 
57 
58 

7482 
7559 
7634 

749° 
7566 
7642 

7497 
7574 
7649 

75°5 
7582 

7657 

75'3 

7589 
7664 

7520 

7597 
7672 

7528 
7604 
7679 

7752 
7825 
7896 

7536 
7612 
7686 

755i 
7627 
7701 

2  2 
2   2 
I   2 

345 
345 
344 

59 
60 
61 

7709 
7782 
7853 

7716 

7789 
7860 

7723 
7796 
7868 

773i 
7803 
7875 

7738 
7810 
7882 

7745 
7818 
7889 

7760  7767 

7832  7839 
7903!  7910 

7774 
7846 
7917 

2 
2 

2 

344 
344 
344 

5  6  7 
5  6  6 
5  6  6 

62 
63 
64 

7924 

7993 
8062 

7931 
8000 
8069 

7938 
8007 
8075 

7945 
8014 
8082 

7952 
8021 
8089 

7959 
8028 
8096 

7966 
8035 
8102 

7973  798o 
8041!  8048 
8109  8116 

7987 
8055 
8122 

2 

2 
2 

334 
334 

334 

5  6  6 
5  5  6 
5  5  6 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176  8182 

8189 

I     2 

334 

5  5  6 

5  5  6 
5  5  6 
4  5  6 

66 
67 
68 

8195 
8261 
8325 

8202 
8267 
8331 

8209 
8274 
8338 

8215 
8280 
8344 

8222 
8287 
8351 

8228 
8293 
8357 

8235 
8299 
8363 

8241 
8306 
8370 

8248 
8312 
8376 

8254 

8319 
8382 

I     2 
I     2 
I     2 

334 

334 
334 

69 
70 

71 

8388 
845  i 
8513 

8395 
8457 
8519 

8401 
8463 
8525 

8407 
8470 
8531 

8414 
8476 
8537 

8420 
8482 
8543 

8426 
8488 
8549 

8432 
8494 
8555 

8439 
8500 
8561 

8445 
8506 
8567 

2 
2 
2 

234 
234 
234 

4  5  6 
4  5  6 

4  5  5 

72 
73 
74 

8573 
8633 
8692 

8579 
8639 
8698 

8585 
8645 
8704 

8591 
8651 
8710 

8597 
8657 
8716 

8603 
8663 
8722 

8609 
8669 
8727 

8615  8621 
8675  8681 
8733  8739 

8627 
8686 
8745 

2 
2 
2 

234 
234 
234 

4  5  5 
4  5  5 
4  5  5 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791  8797 

8802 

2 

233 

455 

4  5  5 
4  4  5 
445 

4  4  5 
4  4  5 
445 

4  4  5 
4  4  5 
445 

76 

77 
78 

8808 
8865 
8921 

8814 
8871 
8927 

8820 
8876 
8932 

8825 
8882 
8938 

8831 
8887 
8943 

8837 
8893 

8949 

8842 
8899 
8954 

8848  8854 
8904  8910 
8960  8965 

8859 
8915 
8971 

2 
2 
2 

233 
233 
233 

79 
80 
81 

8976 
9031 
9085 

8982 
9036 
9090 

8987 
9042 
9096 

8993 
9047 
9101 

8998 

9053 
9106 

9004 
9058 
9112 

9009 
9063 
9117 

9015 
9069 
9122 

9020 
9074 
9128 

9025 
9079 
9133 
9186 
9238 
9289 

2 
2 
2 

233 
233 
233 

82 
83 
84 

9138 
9191 

9243 

9H3 
9196 
9248 

9149 
9201 
9253 

9154 
9206 
9258 

9159 
9212 
9263 

9165 
9217 
9269 

9170 
9222 
9274 

9175 
9227 

9279 

9180 
9232 
9284 

2 
2 
2 

233 
233 
233 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

933° 

9335 

9340 

9390 
9440 
9489 

I     2233 

4  4  5 

86 
87 
88 

9345 
9395 
9445 

9350 
9400 

945° 

9355 
9405 
9455 

9360 
9410 
9460 

9365 
9415 
9465 

937° 
9420 
9469 

9375 
9425 
9474 

938o 
943° 
9479 

9385 
9435 
9484 

I     2 
0     I 
O     I 

233 
223 
223 

4  4  5 
344 
344 

89 
90 

91 

92~ 
93 
94 

9494 
9542 
9590 

9499 
9547 
9595 

95°4 
9552 
9600 

95°9 
9557 
9605 

95*3 
9562 
9609 

95^8 
9566 
9614 

9523 
957i 
9619 

9528 

9576 
9624 

9533 
9581 
9628 

9538 
9586 

9633 

O     I 
O     I 
O     I 

223 
223 
223 

344 
344 
344 

9638 
9685 
9731 

9643 
9689 

9736 

9647 
9694 
974i 

9652 
9699 
9745 

9657 
97°3 
975° 

9661 
9708 
9754 

9666 
9713 
9759 

9671 
9717 
9763 

9675  9680 
9722J  9727 
9768  9773 

O     I 
O     I 
0     I 

223 
223 
223 

344 
344 
344 

95 

9777 

9782 

9*86 

9791 

9795 

9800 

9805 

9809 

9814'  9818 

0     l|  2   2   3 

344 

96 
97 
98 

9823 
9868 
9912 

9827!  9832 
9872  9877 
9917  9921 

9836 
9881 
9926 

9841  9845 
9886  9890 
9930  9934 

9850 
9894 
9939 

9854 
9899 
9943 

9859  9863 
9903  9908 
9948  9952 

0     Ij  2   2   3 

o    1223 
o   1223 

344 
344 
344 

99 

9956 

9961  9965 

9969 

9974  9978J  9983 

9987  9991  9996 

o  i  i  2  2  3J  3  3  4 

282 


NATURAL    SINES. 


0' 

6' 

12' 

18' 

24' 

3O' 

36' 

42' 

48' 

54' 

123 

4  5 

0° 

oooo 

0017 

0035 

0052 

0070 

0087 

0105 

OI22 

0140 

OI57 

369 

12  15 

1 

2 
3 

OI75 
0349 
0523 

0192 
0366 
0541 

0209 
0384 
0558 

0227 
0401 
0576 

0244 
0419 
0593 

0262 
0436 
0610 

0279 

0454 
0628 

0297 
0471 
0645 

0314 
0488 
o663 

0332 
0506 
ob8o 

369 
369 
3  6  9 

12   I5 
12   I5 
12   I5 

4 
5 

6 

0698 
0872 
1045 

0715 
0889 
1063 

0732 
0906 
1080 

0750 
0924 
1097 

0767 
0941 
"15 

0785 
0958 
1132 

0802 
0976 
1149 

0819 

0993 
1167 

0837 

IOII 

1184 

0854 
1028 

I  20  1 

369 
369 
369 

12   I5 

12   I4 
12   14 

7 
8 
9 

1219 
1392 
i564 

1236 
1409 
1582 

1253 
1426 

1599 

1271 
1444 
1616 

1288 
1461 
1633 

1305 
1478 
1650 

1323 

1495 
1668 

1340 
1513 

1685 

1357 
1530 
1702 

1374 
X547 
1719 

369 
369 
369 

12   I4 
12   14 
12   14 

10 

i736 

1754 

1771 

1788 

1805 

1822 

1840 

1857 

1874 

1891 

369 

12   14 

11 
12 
13 

1908 
2079 
2250 

1925 
2096 
2267 

1942 
2113 

2284 

1959 
2130 
2300 

1977 
2147 
2317 

1994 
2164 
2334 

2OII 

2181 

2351 

2028 
2198 
2368 

2045 
2215 
2385 

2062 
2232 
2402 

369 
369 
3  6  8 

II   14 
II   14 

II   14 

14 
15 
16 

2419 
2588 
2756 

2436 
2605 

2773 

2453 
2622 
2790 

2470 
2639 
2807 

2487 
2656 
2823 

2504 
2672 
2840 

2521 
2689 

2857 

2538 
2706 
2874 

2554 
2723 
2890 

257i 
2740 
2907 

368 
3  6  8 
368 

II   14 
II   14 
II   14 

17 
18 
19 

2924 
3090 
3256 

2940 

3107 
3272 

2957 
3I23 
3289 

2974 
3MO 
3305 

2990 

3i56 
3322 

3007 
3i73 
3338 

3024 
3190 

3355 

3040 
3206 
3371 

3057 
3223 
3387 

3°74 
3239 
3404 

368 
3  6  8 
3  5  8 

II   14 
II   14 
II   14 

20 

3420 

3437 

3453 

3469 

3486 

3502 

35i8 

3535 

3551 

3567 

3  5  8 

II   14 

21 
22 
23 

3584 
3746 
3907 

3600 
3762 
3923 

3616 
3778 
3939 

3633 
3795 
3955 

3649 
3811 

397i 

3665 
3827 
3987 

3681 

3843 
4003 

3697 
3859 
4019 

37H 
3875 
4035 

3730 
3891 
405  l 

3  5  8 
3  5  8 
3  5  8 

II   14 

II   14 
II   14 

24 
25 
26 

4067 
4226 
4384 

4083 
4242 
4399 

4099 
4258 
4415 

4"5 

4274 

443  i 

4I31 

4289 
4446 

4H7 
43°5 
4462 

4163 
4321 
4478 

4179 
4337 
4493 

4195 
4352 
45°9 

4210 
4368 
4524 

3  5  8 
3  5  8 
3  5  8 

II   I3 

II   13 

10  13 

27 

28 
29 

4540 

4695 
4848 

4555 
4710 
4863 

4571 
4726 
4879 

4586 
4741 
4894 

4602 

4756 
4909 

4617 
4772 
4924 

4633 
4787 
4939 

4648 
4802 
4955 

4664 
4818 
497° 

4679 
4833 
4985 

3  5  8 
3  5  8 
3  5  8 

10  13 
10  13 
10  13 

30 

5000 

5OI5 

5°3Q 

5°45 

5060 

5°75 

5090 

5I05 

5120 

5135 

3  5  8 

10  13 

31 
32 
33 

5150 
5299 
5446 

5165 
53H 
546i 

5180 
5329 
5476 

5195 
5344 
5490 

5210 

5358 
55°5 

5225 
5373 
5519 

5240 
5388 
5534 

5255 
5402 

5548 

5270 
5417 
5563 

5284 
5432 

5577 

257 

2  5  7 
2  5  7 

10   12 
IO   12 
IO   12 

34 
35 
36 

5592 
5736 
5878 

5606 

575° 
5892 

5621 

5764 
5906 

5635 
5779 
5920 

5650 
5793 
5934 

5664 
5807 
5948 

5678 
5821 
5962 

5693 
5835 
5976 

57°7 
5850 
5990 

572i 
5864 
6004 

2  5  7 
2  5  7 
257 

IO   12 
IO   12 

9  12 

37 

38 
39 

6018 

6l57 
6293 

6032 
6170 
6307 

6046 
6184 
6320 

6060 
6198 
6334 

6074 
6211 
6347 

6088 
6225 
6361 

6101 

6239 
6374 

6115 

6252 
6388 

6129 
6266 
6401 

6i43 
6280 
6414 

2  5  7 
257 
247 

9  12 

9   I' 

9  ii 

40 

6428 

6441 

6455 

6468 

6481 

6494 

6508 

6521 

6534 

6547 

247 

9  ii 

41 
42 
43 

6561 
6691 
6820 

6574 
6704 

6833 

6587 
6717 
6845 

6600 
6730 
6858 

6613 

6743 
6871 

6626 
6756 
6884 

6639 
6769 
6896 

66^2 
6782 

6909 

6665 
6794 
6921 

6678 
6807 
6934 

247 
2  4  6 
246 

9  ii 
9  ii 
8  ii 

44 

|6947 

6959 

6972 

6984 

6997 

7009 

7022 

7°34 

7046 

7°59 

246 

8  10 

NATURAL    SINES. 


283 


0' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48' 

54' 

123 

4  5 

45 

7071 

7083 

7096 

7108 

7120 

7i33 

7H5 

7i57 

7169 

7181 

246 

8  10 

46 
47 
48 

7193 
73H 
743i 

7206 
7325 
7443 

7218 
7337 
7455 

7230 

7349 
7466 

7242 
7361 
7478 

7254 
7373 
7490 

7266 
7385 
7501 

7278 
7396 
75J3 

7290 
7408 
7524 

7302 
7420 
7536 

2  4  6 
2  4  6 
246 

8  10 
8  10 
8  10 

49 
50 
51 

7547 
7660 
7771 

7558 
7672 
7782 

757° 
7683 
7793 

758i 
7694 
7804 

7593 
7705 
7815 

7604 
7716 
7826 

7615 

7727 

7837 

7627 
7738 
7848 

7638 
7749 
7859 

7649 
7760 
7869 

246 
2  4  6 
2  4  5 

8   9 
7   9 
7   9 

52 
53 
54 

7880 
7986 
8090 

7891 

7997 
8100 

7902 
8007 
8111 

7912 
8018 
8121 

7923 
8028 
8131 

7934 
8039 
8141 

7944 
8049 
8151 

7955 
8059 
8161 

7965 
8070 
8171 

7976 
8080 
8181 

245 
235 
2  3  5 

7   9 
7   9 
7   8 

55 

8192 

8202 

8211 

8221 

8231 

8241 

8251 

8261 

8271 

8281 

2  3  5 

7   8 

56 

57 
58 

8290 

8387 
8480 

8300 
8396 
8490 

8310 
8406 
8499 

8320 

8415 
8508 

8329 
8425 
8517 

8339 
8434 
8526 

8348 
8443 
8536 

8358 
8453 
8545 

8368 
8462 
8554 

8377 
8471 
8563 

2  3  5 
2  3  5 
235 

6   8 
6   8 
6   8 

59 
60 
61 

8572 
8660 
8746 

8581 
8669 
8755 

8590 
8678 
8763 

8599 
8686 
8771 

8607 
8695 
8780 

8616 
8704 
8788 

8625 
8712 
8796 

8634 
8721 
8805 

8643 

8729 
8813 

8652 
8738 
8821 

3  4 

3  4 
3  4 

6   7 
6   7 
6   7 

62 
63 
64 

8829 
8910 
8988 

8838 
8918 
8996 

8846 
8926 
9003 

8854 
8934 
9011 

8862 
8942 
9018 

8870 
8949 
9026 

8878 
8957 
9033 

8886 
8965 
9041 

8894 

8973 
9048 

8902 
8980 
9056 

3  4 
3  4 
3  4 

1  1 

5   6 

65 

9063 

9070 

9078 

9085 

9092 

9100 

9107 

9114 

9121 

9128 

2  4 

5   6 

66 
67 
68 

9135 
9205 
9272 

9M3 
9212 

9278 

915° 
9219 
9285 

9157 
9225 
9291 

9164 
9232 
9298 

9171 
9239 
9304 

9178 
9245 
93" 

9184 
9252 
9317 

9191 
9259 
9323 

9198 
9265 
9330 

2  3 
2  3 
2  3 

5   6 

4   6 
4   5 

69 
70 
71 

9336 
9397 
9455 

9342 
9403 
9461 

9348 
9409 
9466 

9354 
94i5 
9472 

936i 
9421 
9478 

9367 
9426 

9483 

9373 
9432 
9489 

9379 
9438 
9494 

9385 
9444 
9500 

9391 
9449 
9505 

2  3 
2  3 
2  3 

4   5 
4   5 
4   5 

72 
73 
74 

9511 
9563 
9613 

95l6 
9568 
9617 

952i 
9573 
9622 

9527 
9578 
9627 

9532 
9583 
9632 

9537 
9588 
9636 

9542 
9593 
9641 

9548 
9598 
9646 

9553 
9603 
9650 

9558 
9608 

9655 

2  3 

2   2 
2   2 

4   4 
3   4 
3   4 

75 

9659 

9664 

9668 

9673 

9677 

9681 

9686 

9690 

9694 

9699 

I   2 

3   4 

76 

77 
78 

97°3 
9744 
978i 

9707 
9748 
9785 

9711 
9751 
9789 

97'5 
9755 
9792 

9720 

9759 
9796 

9724 
9763 
9799 

9728 
9767 
9803 

9732 
9770 
9806 

9736 
9774 
9810 

9740 
9778 
98i3 

2 
2 
I      2 

3   3 
3   3 
2   3 

79 
80 
81 

9816 
9848 
9877 

9820 
9851 
9880 

9823 

9854 
9882 

9826 

9857 
9885 

9829 
9860 
9888 

9833 
9863 
9890 

9836 
9866 
9893 

9839 
9869 

9895 

9842 
9871 
9898 

9845 
9874 
9900 

I      2 
O 
0 

2   3 

2    2 
2    2 

82 
83 
84 

9903 
9925 
9945 

9905 
9928 

9947 

9907 
993° 
9949 

9910 
9932 

995  i 

9912 
9934 
9952 

9914 
9936 
9954 

9917 
9938 
9956 

9919 
9940 
9957 

9921 
9942 
9959 

9923 
9943 
9960 

O 
0 
O 

2    2 
I    2 
I    I 

85 

9962 

9963 

9965 

9966 

9968 

9969 

9971 

9972 

9973 

9974 

001 

I    I 

86 
87 
88 

9976 
9986 
9994 

9977 
9987 
9995 

9978 
9988 
9995 

9979 
9989 
9996 

9980 
9990 
9996 

9981 
9990 
9997 

9982 
9991 
9997 

9983 
9992 
9997 

9984 
9993 
9998 

9935 
9993 
9998 

O   O   I 
000 
O   O   O 

I    I 

O    O 

89 

9998 

9999 

9999 

9999 

9999 

rooo 

nearly. 

rooo 

nearly. 

rooo 

nearly. 

rooo 

nearly. 

rooo 

nearly. 

O   O   O 

O    O 

284 


NATURAL   COSINES. 


0' 

6' 

12' 

18' 

24' 

3O' 

36' 

42' 

48' 

54' 

123 

4  5 

0° 

I  '000 

I  '000 

nearly. 

rooo 

nearly. 

rooo 

nearly. 

I'OOO 
nearly. 

9999 

9999 

9999 

9999 

9999 

000 

o   o 

1 

2 
3 

9998 

9994 
9986 

9998 

9993 
9985 

9998 

9993 
9984 

9997 
9992 

9983 

9997 
9991 
9982 

9997 
9990 
9981 

9996 
9990 
9980 

9996 
9989 
9979 

9995 
9988 
9978 

9995 
9987 

9977 

o  o  o 
o  o  o 

O   O   I 

o   o 
I   I 
I   I 

4 
5 
6 

9976 
9962 
9945 

9974 
9960 

9943 

9973 
9959 
9942 

9972 

9957 
9940 

9971 

9956 
9938 

9969 
9954 
9936 

9968 
9952 
9934 

9966 

995  i 
9932 

9965 
9949 
9930 

9963 
9947 
9928 

o  o 

O   I 
0   I 

I   I 

I    2 
I    2 

7 
8 
9 

9925 
9903 
9877 

9923 
9900 

9874 

9921 
9898 
9871 

9919 

9895 
9869 

9917 

9893 
9866 

9914 
9890 
9863 

9912 
9888 
9860 

9910 

9885 
9857 

-9907 
9882 
9854 

9905 
9880 

9851 

O   I 
O   I 
0   I 

2    2 
2    2 
2    2 

10 

9848 

9845 

9842 

9839 

9836 

9833 

9829 

9826 

9823 

9820 

112 

2    3 

11 
12 
13 

9816 
9781 
9744 

9813 

9778 
9740 

9810 
9774 
9736 

9806 
9770 
9732 

9803 
9767 
9728 

9799 
9763 
9724 

9796 

9759 
9720 

9792 
9755 
97J5 

9789 
9751 
9711 

9785 
9748 
9707 

I   I   2 
I   I   2 
I   I   2 

2    3 

3   3 
3   3 

14 
15 
16 

97°3 
9659 
9613 

9699 

9655 
9608 

9694 
9650 
9603 

9690 
9646 
9598 

9686 
9641 
9593 

9681 
9636 
9588 

9677 
9632 
9583 

9673 
9627 

9578 

9668 
9622 
9573 

9664 
9617 
9568 

I   2 
2   2 
2   2 

3   4 
3   4 
3   4 

17 
18 
19 

9563 
9511 
9455 

9558 
95°5 
9449 

9553 
9500 

9444 

9548 
9494 
9438 

9542 
9489 
9432 

9537 
9483 
9426 

9532 
9478 
942i 

9527 
9472 
9415 

952i 
9466 
9409 

95l6 
9461 

9403 

2   3 
2   3 
2   3 

4   4 
4   5 
4   5 

20 

9397 

939i 

9385 

9379 

9373 

9367 

936i 

9354 

9348 

9342 

2   3 

4   5 

21 
22 
23 

9336 
9272 
9205 

9330 
9265 
9198 

9323 
9259 
9191 

9317 
9252 
9184 

93U 
9245 
9178 

9304 
9239 
9171 

9298 
9232 
9164 

9291 
9225 
9157 

9285 
9219 

915° 

9278 
9212 

9H3 

2   3 
2   3 
2   3 

4   I 
4   6 

5   6 

24 
25 
26 

9135 
9063 
8988 

9128 
9056 
8980 

9121 
9048 
8973 

9114 
9041 
8965 

9107 
9033 
8957 

9100 
9026 
8949 

9092 
9018 
8942 

9085 
9011 
8934 

9078 
9003 
8926 

9070 
8996 
8918 

I   2   4 

i  3  4 
i  3  4 

5   6 
5   6 
5   6 

27 
28 
29 

8910 
8829 
8746 

8902 
8821 
8738 

8894 
8813 
8729 

8886 
8805 
8721 

8878 
8796 
8712 

8870 
8788 
8704 

8862 
8780 
8695 

8854 
8771 
8686 

8846 

8763 
8678 

8838 

8755 
8669 

i  3  4 
i  3  4 
i  3  4 

5   7 
6   ? 

30 

8660 

8652 

8643 

8634 

8625 

8616 

8607 

8599 

8590 

8581 

i  3  4 

6   7 

31 
32 

33 

8572 
8480 
8387 

8563 
8471 

8377 

8554 
8462 
8368 

8545 
8453 
8358 

8536 
8443 
8348 

8526 
8434 
8339 

8517 
8425 
8329 

8508 

8415 
8320 

8499 
8406 
8310 

8490 
8396 
8300 

2  3  5 
235 
2  3  5 

6   8 
6   8 
6   8 

34 
35 
36 

8290 
8192 
8090 

8281 
8181 
8080 

8271 
8171 
8070 

8261 
8161 
8059 

8251 
8151 
8049 

8241 
8141 
8039 

8231 
8131 
8028 

8221 
8121 
8018 

8211 
8111 
8007 

8202 
8100 
7997 

2  3  5 
235 
2  3  5 

7   8 
7   8 
7   9 

37 
38 

39 

7986 
7880 
7771 

7976 
7869 
7760 

7965 
7859 
7749 

7955 
7848 

7738 

7944 
7837 
7727 

7934 
7826 
7716 

7923 
7815 

77°5 

7912 
7804 
7694 

7902 
7793 
7683 

7891 
7782 
7672 

245 
245 
246 

7   9 
7   9 
7   9 

40 

7660 

7649 

7638 

7627 

76i5 

7604 

7593 

758i 

757° 

7559 

2  4  6 

8   9 

41 
42 
43 

7547 
743i 
73H 

7536 
7420 
7302 

7524 
7408 
7290 

75'3 
7396 
7278 

7501 

™ll 
7266 

7490 
7373 
7254 

7478 
736i 
7242 

7466 

7349 
7230 

7455 
7337 
7218 

7443 
7325 
7206 

2  4  6 
2  4  6 
246 

8  10 
8  10 
8  10 

44 

7J93 

7181 

7169 

7i57 

7H5 

7133 

7120 

7108 

7096 

7083 

246 

8  10 

N.B.  —  Numbers  in  difference-columns  to  be  subtracted,  not  added. 


NATURAL   COSINES. 


28S 


0' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48'  54' 

123 

4  5 

45° 

7071 

7059 

7046 

7034 

7022 

7009 

6997 

6984 

6972 

6959 

2  4  6 

8  10 

46 
47 
48 

IsT 

50 
51 

6947 
6820 
6691 

6934 
6807 
6678 

6921 
6794 
6665 

6909 
6782 
6652 

6896 
6769 
6639 

6884 
6756 
6626 

6871 

6743 
6613 

6858 
6730 
6600 

6845 
6717 
6587 

6833 
6704 
6574 

2  4  6 
2  4  6 
247 

8  ii 
9  ii 
9  ii 

6561 
6428 
6293 

6547 
6414 
6280 

6534 
^401 
6266 

6521 
6388 
6252 

6508 

6374 
6239 

6494 
6361 
6225 

6481 

6347 
6211 

6468 

6334 
6198 

6455 
6320 
6184 

6441 
6307 
6170 

2  4  7 
247 
2  5  7 

9  ii 
9  ii 
9  ii 

52 
53 
54 

6l57 
6018 
5878 

6i43 
6004 

5864 

6129 
5990 
5850 

6115 
5976 
5835 

6101 
5962 
5821 

6088 
5948 
5807 

6074 
5934 
5793 

6060 
5920 
5779 

6046 
5906 
5764 

6032 
5892 
575° 

257 
257 
2  5  7 

9  12 
9  12 
9  12 

55 

le" 

57 
58 

5736 

572i 

57°7 

5693 

5678 

5664 

5650 

5635 

5621 

5606 

257 

10   12 

5592 
5446 
5299 

5577 
5432 
5284 

5563 
54i7 
5270 

5548 
5402 

5255 

5534 
5388 
5240 

55i9 

5373 
5225 

5505 
5358 
5210 

5490 
5344 
5195 

5476 
5329 
5180 

546i 
53H 
5165 

257 
257 
257 

IO   12 
10   12 
10   12 

59 
60 
61 

5150 
5000 
4848 

5135 

4985 
4833 

5120 

497° 
4818 

5io5 
4955 
4802 

5090 
4939 
4787 

5075 
4924 
4772 

5060 
4909 
4756 

5045 
4894 
4741 

5030 

4879 
4726 

5015 
4863 
4710 

3  5  8 
3  5  8 
3  5  8 

10  13 
10  13 
10  13 

62 
63 
64 

4695 
4540 
4384 

4679 

4524 
4368 

4664 
45°9 
4352 

4648 
4493 
4337 

4633 
4478 
4321 

4617 
4462 
4305 

4602 
4446 
4289 

4586 
443i 
4274 

457i 
4415 
4258 

4555 
4399 
4242 

3  5  8 
3  5  8 
3  5  8 

10  13 
10  13 

II  13 

65 

4226 

4210 

4195 

4179 

4163 

4H7 

4131 

4H5 

4099 

4083 

3  5  8 

II  13 

66 
67 
68 

4067 
3907 
3746 

4051 
3891 
3730 

4035 
3875 
37  H 

4019 

3859 
3697 

4003 

3843 
3681 

3987 

3??7 
3665 

3971 
3811 
3649 

3955 
3795 
3633 

3939 
3778 
3616 

3923 
3762 
3600 

3  5  8 
3  5  8 
3  5  8 

II  14 

II  14 
II  14 

69 
70 
71 

~72~ 
73 
74 

3584 
3420 
3256 

3567 
3404 
3239 

355i 
3387 
3223 

3535 
3371 
3206 

35i8 

3355 
3190 

35°2 
3338 
3173 

3486 
3322 
3*56 

3469 
3305 
3MO 

3453 
3289 
3123 

3437 
3272 
3107 

3  5  8 
3  5  8 
368 

II  14 

II  14 
II  14 

3090 

2924 
2756 

3074 
2907 
2740 

3057 
2890 

2723 

3040 
2874 
2706 

3024 

2857 
2689 

300,7 
2840 
2672 

2990 
2823 
2656 

2974 
2807 
2639 

2957 
2790 
2622 

2940 

2773 
2605 

368 
368 
3  6  8 

II  14 

II  14 

II  .14 

75 

2588 

2571 

2554 

2538 

2521 

2504 

2487 

2470 

2453 

2436 

368 

II  14 

76 
77 
78 

2419 
2250 
2079 

2402 

2233 
2062 

2385 
2215 
2045 

2368 
2198 
2028 

235  l 
2181 

2OII 

2334 
2164 
1994 

2317 
2147 
1977 

2300 
2130 
1959 

2284 

2113 

1942 

2267 
2096 
1925 

368 
369 
369 

II  14 
II  14 

II  14 

79 
80 
81 

~S2 
83 
84 

1908 
1736 
i564 

1891 
1719 

1547 

1874 
1702 
1530 

1857 
1685 

I5U 

1840 
1668 
H95 

1822 
1650 
1478 

1805 

1633 
1461 

1788 
1616 
1444 

1771 

1599 
1426 

1754 
1582 
1409 

369 
369 
369 

12   I4 
12   14 

12   14 

1392 
1219 
1045 

1374 

I2OI 
1028 

1357 
1184 

IOII 

1340 
1167 

0993 

1323 
1149 
0976 

1305 
1132 

0958 

1288 

"i5 

0941 

1271 
1097 
0924 

1253 
1080 
0906 

1236 
1063 
0889 

369 
369 
369 

12   I4 
12   14 
12   I4 

85 

0872 

0854 

0837 

0819 

0802 

0785 

0767 

0750 

0732 

0715 

369 

12   I5 

86 
87 
88 

0698 
0523 
0349 

0680 
0506 
0332 

o663 
0488 
03H 

0645 
0471 
0297 

0628 

°454 
0279 

0610 
0436 
0262 

°593 
0419 
0244 

0576 
0401 
0227 

0558 
0384 
0209 

0541 
0366 
0192 

369 
369 
369 

12   15 
12   I5 
12   I5 

89 

oi75 

0157 

0140 

OI22 

0105 

0087 

0070 

0052 

0035 

0017 

369 

12   I5 

N.B.  —  Numbers  in  difference-columns  to  be  subtracted,  not  added. 


286 


NATURAL   TANGENTS. 


0' 

& 

12' 

18' 

24' 

3O' 

36' 

42' 

48' 

54' 

123 

4  5 

0° 

•oooo 

0017 

0035 

0052 

0070 

0087 

0105 

OI22 

0140 

OI57 

369 

12  14 

1 

2 
3 

•oi75 
•0349 
•0524 

0192 
0367 
0542 

0209 
0384 
0559 

0227 
0402 
°577 

0244 
0419 
0594 

0262 

°437 
0612 

0279 

0454 
0629 

0297 
0472 
0647 

03H 

0489 
0664 

0332 
0507 
0682 

369 
369 
369 

12  15 
12  15 
12  I5 

4 
5 
B 

•0699 
•0875 
•1051 

0717 
0892 
1069 

0734 
0910 
1086 

0752 
0928 
1104 

0769 
0945 

1122 

0787 
0963 
H39 

0805 
0981 
"57 

0822 
0998 

"75 

0840 
1016 
1192 

0857 
103? 

I2IO 

369 
369 
369 

12  I5 
12  I5 
12  I5 

7 
8 
9 

•1228 
•1405 
•1584 

1246 

1423 
1602 

1263 
1441 
1620 

1281 

H59 
1638 

1299 

1477 
1655 

1317 

H95 
1673 

1334 
1512 
1691 

1352 
1530 
1709 

1370 
1548 
1727 

1388 
1566 

1745 

369 
369 
369 

12  I5 
12  15 
12  I5 

10 

•1763 

1781 

1799 

1817 

1835 

1853 

1871 

1890 

1908 

1926 

369 

12  I5 

11 
12 
13 

•1944 
•2126 
•2309 

1962 
2144 
2327 

1980 
2162 
2345 

1998 
2180 
2364 

2016 
2199 
2382 

2035 
2217 
2401 

2053 

2235 
2419 

2071 

2254 
2438 

2089 
2272 
2456 

2107 
2290 
2475 

369 
369 
369 

12  I5 
12  I5 
12  I5 

14 
15 
16 

•2493 
•2679 
•2867 

2512 
2698 
2886 

2530 
2717 
2905 

2549 
2736 
2924 

2568 
2754 
2943 

2586 

2773 
2962 

2605 
2792 
2981 

2623 
2811 
3000 

2642 
2830 
3019 

2661 
2849 
3038 

369 
369 
369 

12  l6 

13  J6 
13  16 

17 
18 
19 

•3057 
•3249 
'3443 

3076 
3269 
3463 

3096 
3288 
3482 

3"5 

3307 
3502 

3134 
3327 
3522 

3153 
3346 
3541 

3172 
3365 
356i 

3i9i 
3385 

358i 

3211 

3404 
3600 

3230 
3424 
3620 

3  6  10 

3  6  10 
3  6  10 

13  16 
13  16 
13  17 

20 

•3640 

3659 

3679 

3699 

3719 

3739 

3759 

3779 

3799 

3819 

3  7  10 

13  17 

21 
22 
23 

•3839 
•4040 

•4245 

3859 
4061 
4265 

3879 
4081 
4286 

3899 
4101 

43°7 

3919 
4122 

4327 

3939 
4142 
4348 

3959 
4163 
4369 

3979 
4183 
4390 

4000 
4204 
4411 

4020 
4224 
443  i 

3  7  I0 
3  7  I0 
3  7  I0 

13  17 

H  17 
14  17 

24 
25 
26 

•4452 
•4663 
•4877 

4684 
4899 

4494 
4706 
4921 

45i5 

4727 

4942 

4536 
4748 
4964 

4557 
4770 
4986 

4578 
4791 
5008 

4599 
4813 
5029 

4621 
4834 
5051 

4642 
4856 
5°73 

4  7  10 
4  7  ii 
4  7  ii 

14  18 
14  18 
15  18 

27 

28 
29 

•5095 
•5317 
'5543 

5"7 

5340 
5566 

5139 
5362 

5589 

5161 

5384 
5612 

5184 
5407 
5635 

5206 

5430 
5658 

5228 
5452 
5681 

525° 

5475 
5704 

5272 
5498 
5727 

5295 
5520 

5750 

4  7  ii 

4  8  ii 
4  8  12 

15  18 
15  19 
15  19 

30 

'5774 

5797 

5820 

5844 

5867 

5890 

59H 

5938 

596i 

5985 

4  8  12 

16  20 

31 
32 
33 

•6009 
•6249 
•6494 

6032 
6273 
6519 

6056 
6297 
6544 

6080 
6322 
6569 

6104 
6346 
6594 

6128 

6371 
6619 

6152 

6395 
6644 

6176 
6420 
6669 

6200 

6445 
6694 

6224 
6469 
6720 

4  8  12 
4  8  12 
4  8  13 

16  20 
16  20 

I7  21 

34 
35 
36 

•6745 
•7002 
•7265 

6771 
7028 
7292 

6796 
7°54 
7319 

6822 
7080 
7346 

6847 
7107 

7373 

6873 
7!33 
7400 

6899 

7J59 
7427 

6924 
7186 

7454 

6950 
7212 
7481 

6976 

7239 
7508 

4  9  13 
4  9  13 
5  9  H 

17  21 
18  22 

18  23 

37 
38 
39 

7536 
7813 
•8098 

7563 
7841 
8127 

7590 
8156 

7618 
7898 
8185 

7646 
7926 
8214 

7673 
7954 
8243 

7701 
7983 
8273 

7729 
8012 

8302 

7757 
8040 

8332 

7785 
8069 
8361 

5  9  H 
5  I0  M 
5  10  15 

18  23 

19  24 

20  24 

40 

•8391 

8421 

8451 

8481 

8511 

8541 

857i 

8601 

8632 

8662 

5  I0  J5 

20  25 

41 
42 
43 

•8693 
•9004 
•9325 

8724 
9036 
9358 

8754 
9067 

939i 

8785 
9099 
9424 

8816 
9131 
9457 

8847 
9163 
9490 

8878 
9195 
9523 

8910 
9228 
9556 

8941 
9260 
9590 

8972 
9293 
9623 

5  10.  16 

5  ii  16 
6  ii  17 

21  26 
21  27 
22  28 

44 

,9657 

9691 

9725 

9759 

9793 

9827 

9861 

9896 

993° 

9965 

6  ii  17 

23  29 

NATURAL   TANGENTS. 


287 


45° 

47 
48 

0' 

6'  12'  18' 

24'  '  30' 

36' 

42' 

48' 

54' 

123 

4  5 

roooo 

0035  0070;  0105 

0141  0176 

O2  1  2 

0247 

0283 

0319 

6  12  18 

24  30 

1-0724 
1-1106 

0392 
0761 
"45 

0428  0464 
0799  0837 
1184  1224 

0501 

0875 
1263 

0538 
0913 

1303 

0575 
095  ! 
1343 

0612 
0990 
1383 

0649 
1028 
1423 

0686 
1067 
1463 

6  12  18 

6  13  19 
7  13  20 

25  31 
25  32 
26  33 

49 
50 
51 

1-1504 
1-1918 
1-2349 

1544 
1960 

2393 

1585  1626 

2OO2!  2045 
2437  2482 

1667 
2088 
2527 

1708 
2131 

2572 

1750 
2174 
2617 

1792 
2218 
2662 

1833 
2261 
2708 

1875 
2305 

2753 

7  14  21 

7   14   22 

8  15  23 

28  34 
29  36 
30  38 

52 
53  , 

54  1 

1-2799 
1-3270 
'I-3764 

2846 
33*9 
3814 

2892 
3367 
3865 

2938 
3416 
3916 

2985 
3465 
3968 

3032 
35'4 
4019 

3079 
3564 
4071 

3127 

3613 
4124 

4176 

3222 
4229 

8  16  23 
8  16  25 
9  17  26 

3i  39 
33  4i 
34  43 

55 

1-4281 

4335 

4388 

4442 

4496 

4550 

4605 

4659 

47i5 

4770 

9  18  27 

36  45 

56 
57 
58 

1-4826 

1-5399 
1-6003 

4882  4938  4994 

5458  5517!  5577 
6066  6128  6191 

5051 
5637 
6255 

5108 

5697 
6319 

5l66 

5757 
6383 

5224 
5818 

6447 

5282 
5880 
6512 

5340 
6577 

10  19  29 

10   20   30 
II   21   32 

38  48 
40  50 

43  53 

59 
60 
61 

1-6643 
1-7321 
1-8040 

6709;  6775 
7391  7461 
8115  8190 

6842 

7532 
8265 

6909  6977 

7603!  7675 
8341!  8418 

7°45 
7747 
8495 

7"3 

7820 

8572 

7182 

7893 
8650 

725  i 
7966 
8728 

"  23  34 

12   24   36 
13   26   38 

45  56 
48  60 
51  64 

62 
63 
64 

1-8807 
1-9626 
2-0503 

8887  8967 

97"  9797 
0594  0686 

9047 
9883 
0778 

9128 
9970 
0872 

9210 
0057 
0965 

9292 
0145 
1060 

9375 
0233 
"55 

945s 

0323 
1251 

9542 
0413 

1348 

14   27   41 

15  29  44 
16  31  47 

55  68 
58  73 
63  78 

65 

2-1445 

1543!  1642 

1742 

1842 

1943 

2045 

2148 

2251 

2355 

17  34  51 

68  85 

66 
67 
68 

2-2460 
2-3559 
2-475  i 

2566  2673 
3673  3789 
4876  5002 

2781 
3906 
5129 

2889 
4023 

5257 

2998 
4142 
5386 

3109 
4262 

5517 

3220 
4383 
5649 

3332 
45°4 
5782 

3445 
4627 

18  37  55 

20   40   60 

22  43  65 

74  92 
79  99 
87  108 

69 
70 

71 

2-605  l 
2-7475 
2-9042 

6187  6325 
7625|  7776 
92o8j  9375 

6464 
7929 
9544 

6605 
8083 

97J4 

6746 
8239 
9887 

6889 
8397 
0061 

7°34 
8556 

0237 

7179 
8716 

0415 

7326 
8878 

°595 

24  47  7i 
26  52  78 

29  58  87 

95  "8 
104  130 

"5  H4 

72 

73 

74 

3-0777 
3-2709 

J4874 

0961 
2914 
5105 

1146 
3122 

5339 

3332 
5576 

3544 
5816 

1716 

3759 
6059 

1910 

3977 
6305 

2106 
6554 

2305 
4420 
6806 

2506 
4646 
7062 

32  64  96 
36  72  108 

41   82  122 

129  161 
144  i  80 
162  203 

75 

3-732I 

7583 

7848  8118 

8391 

8667 

8947 

9232 

9520 

9812 

46  94  139 

186  232 

76 

77 
78 

4-0108 
4-33I5 
4-7046 

0408 
3662 

7453 

0713 
4015 

7867 

IO22 

4374 
8288 

1335 

4737 
8716 

1653 
5107 
9152 

1976 
5483 
9594 

2303 
5864 
0045 

2635 
6252 

0504 

2972 
6646 
0970 

53  107  160 
62  124  186 
73  146  219 

214  267 
248  310 

292  365 

79 
80 
81 

5^446 
5-67I3 
6-3138 

1929 
7297 
3859 

2422 
7894 
4596 

2924 
8502 
5350 

3435 
9124 
6122 

3955 
9758 
6912 

4486 
0405 
7920 

5026 
1066 
8548 

5578 
1742 

9395 

6140 
2432 
0264 

87  175  262 

35°  437 

Difference-columns 
cease  to  be  useful,  owing 
to  the  rapidity  with 
which  the  value  of  the 
tangent  changes. 

82 
83 
84 

7'"54 
8-1443 

2066 
2636 
9-677 

3002 
3863 
9-845 

3962 
5126 

IO-O2 

4947 
6427 

IO'2O 

5958 

7769 
10-39 

6996 

9152 
10-58 

8062 

0579 
10-78 

9158 
2052 
10-99 

0285 

3572 
1  1  -20 

85 

ii-43 

i'-66 

11-91 

12-16 

12-43 

12-71  13-00 

13-30 

13-62 

I3-95 

86 
87 
88 

14-30 
19-08 
28-64 

14-67 
I9-74 
30-14 

15-06 
20-45 
31-82 

15-46 
2I'2O 
3J69 

I5-89 
22-O2 
35-80 

16-35  J6'83 
22-90  23-86 
38-19140-92 

17-34 
24-90 
44-07 

17-89 
26-03 
47-74 

18-46 
27-27 
52-08 

89 

57-29 

63-66 

7I  -62 

81-85 

95'49 

114-6  143-2 

191-0 

286-5 

573-o 

! 


288 


NATURAL   COTANGENTS. 


0' 

6' 

12' 

18' 

24'  3O'  36' 

42' 

48'  54' 

Difference-columns 
not  useful  here,  owing 
to  the  rapidity  with 
which  the  value  of  the 
cotangent  changes. 

0° 

Inf. 

573-o 

286-5 

191-0 

143-2 

114-695-49 

81-85 

71-6263-66 

1 

2 
3 

57-29 
28-64 
19-08 

52-08 

27-27 
18-46 

47-74 
26-03 
17-89 

44-07 
24-90 

17-34 

40-92 
23-86 
16-83 

38-I9 
22-90 

16-35 

35-80 
22-02 
15-89 

33-69 

2  1  -2O 
I5-46 

31-82 

20-45 
15-06 

30-I4 
'19-74 
14-67 

4 
5 
6 

14-30 
"'43 
9"5J44 

I3-95 

11-20 

3572 

13-62 
10-99 
2052 

iJ30 
10-78 

°579 

13-00 
10-58 

.9152 

I2-7I 
10-39 

7769 

12-43 

IO'2O 
6427 

12-16 

IO'O2 
5126 

11-91  n-66 
9-845  9-677 
3863  2636 

7 
8 
9 

8-1443 
7-ii54 
6-3138 

0285 
0264 
2432 

9158 

9395 
1742 

8062 
8548 
1066 

6996 
7920 
0405 

5958 
6912 

9758 

4947 
6122 
9124 

3962 

535° 
8502 

3002 
4596 
7894 

2066 
3859 
7297 

10 

5'67i3 

6140 

5578 

5026 

4486 

3955  3435 

2924 

2422 

1929 

123 

4  5 

11 
12 
13 

5-I446 
4-7046 

4-33I5 

0970 
6646 
2972 

0504 
6252 
2635 

0045 
5864 
2303 

9594 
5483 
1976 

9152 
5107 
1653 

8716 
4737 
1335 

8288 
4374 

IO22 

7867 
4015 
0713 

7453 
3662 
0408 

74  148  222 

63  125  188 
53  107  160 

296  370 
252  314 
214  267 

14 
15 
16 

4-0108 
37321 
3'4874 

9812 
7062 
4646 

9520 
6806 
4420 

9232 

6554 
4197 

8947  8667 
6305  6059 
3977  3759 

839i 
5816 

3544 

8118 
5576 
3332 

7848 
5339 
3122 

7583 
5105 
2914 

46  93  J39 

41   82  122 
36   72  I  08 

i  86  232 
163  204 
144  180 

17 
18 
19 

372709 

3-0777 
.  2-9042 

2506 

°595 
8878 

2305 

0415 
8716 

2106 
0237 
8556 

1910 
0061 
8397 

1716 
9887 
8239 

£524 

97H 
8083 

^334 

9544 
7929 

1146  0961 
9375!  9208 
7776!  7625 

32   64   96 

29   58   87 
26   52   78 

129  161 

115  J44 
104  130 

20 

27475 

7326 

7179 

7°34 

6889 

6746 

6605 

6464 

63251  6187 

24  47  71 

95  n8 

21 
22 
23 

2-6051 

2'475  I 

2-3559 

5916 
4627 

3445 

5782 
45°4 
3332 

5649 
4383 
3220 

55!7 
4262 
3109 

5386 
4142 
2998 

5257 
4023 
2889 

5I29 
3906 
2781 

5002  4876 

3789  3673 
2673  2566 

22  43  65 
20  40  60 
18  37  55 

87  108 
79  99 
74  92 

24 
25 
26 

'2-2460 
2-1445 
2-0503 

2355 
1348 

0413 

2251 
1251 

0323 

2148 
"55 
0233 

2045 
1060 

0145 

1943 
0965 
0057 

1842 
0872 
9970 

1742 
0778 
9883 

1642 
0686 

9797 

1543 
0594 
9711 

17  34  51 
16  31  47 
15  29  44 

68  85 
63  78 

58  73 

27 
28 
29 

1-9626 
1-8807 
1-8040 

9542 
8728 
7966 

9458 
8650 

7893 

9375 
8572 
7820 

9292 
8495 
7747 

9210 
8418 
7675 

9128 
8341 
7603 

9047 
8265 
7532 

8967 
8190 

746i 

8887 
8115 
739i 

14  27  41 
1  3  26  38 

12   24   36 

55  68 
51  64 
48  60 

30 

17321 

7251 

7182 

7"3 

7°45 

6977 

6909 

6842 

6775 

6709 

ii  23  34 

45  56 

31 
32 
33 

~3T 
35 
36 

1-6643 
1-6003 
1-5399 

6577 
594i 
5340 

6512 
5880 
5282 

6447 
5818 

5224 

6383 

5757 
5166 

6319 

5697 
5108 

6255 
5637 
5051 

6191 

5577 
4994 

6128 

5517 
4938 

6066 

5458 
4882 

II   21   32 
10   20   30 

10  19  29 

43  53 
40  5° 
38  48 

1-4826 
1-4281 
1-3764 

477° 
4229 

3713 

4715 
4176 

3663 

4659 
4124 

3613 

4605 
4071 
3564 

4550 
4019 

35H 

4496 
3968 
3465 

4442 
3916 
34i6 

4388 
3865 
3367 

4335 
3814 
3319 

9  18  27 
9  17  26 
8  16  25 

36  45 
34  43 
33  4i 

37 
38 
39 

1-3270 
1-2799 
1-2349 

3222 

2753 
2305 

3175 
2708 
2261 

3127 
2662 
2218 

3079 
2617 
2174 

3032 
2572 
2131 

2985 
2527 
2088 

2938 
2482 
2045 

2892 
2437 

2OO2 

2846 

2393 
1960 

8  16  23 
8  15  23 

7   14   22 

3i  39 

30  38 
29  36 

40 

1-1918 

1875 

1833 

1792 

'75° 

1708 

1667 

1626 

1585 

1544 

7  '4  21 

28  34 

41 
42 
43 

1-1504 
1-1106 
1-0724 

1463 
1067 
0686 

H23 
1028 
0649 

1383 
0990 
0612 

1343 

095  i 
0575 

1303 
0913 

0538 

1263 
0875 
0501 

1224 

0837 
0464 

1184 

0799 
0428 

"45 
0761 
0392 

7  13  20 
6  13  19 
6  12  18 

26  33 
25  32 

25  3i 

44 

I  !-°355 

0319 

0283 

0247 

O2  1  2 

0176 

0141 

0105 

OO7O 

0035 

6  12  18 

24  30 

N.  B.  —  Number 


difference-columns  to  be  subtracted,  not  added. 


NATURAL   COTANGENTS. 


289 


45° 

0' 

6' 

12     18' 

24' 

3O' 

36' 

42' 

48' 

54' 

123 

4  5 

ro 

0-9965 

0-9930  0-9896 

0-9861 

0-9827 

0-9793 

o'9759 

0-9725 

0-9691 

6  ii  17 

23  29 

46 
47 
48 

~49~ 
50 
51 

•9657 
•9325 
•9004 

9623 
9293 
8972 

9590 
9260 
8941 

9556 
9228 
8910 

9523 
9195 
8878 

9490 
9163 
8847 

9457 
9131 
8816 

9424 
9099 
8785 

939i 
9067 

8754 

9358 
9036 
8724 

6  ii  17 

5  ii  16 
5  i°  16 

22  28 
21  27 
21  26 

•8693 
•8391 
•8098 

8662 
8361 
8069 

8632 

8332 
8040 

8601 
8302 
8012 

8571 
8273 
7983 

8541 
8243 

7954 

8511 
8214 
7926 

8481 
8185 
7898 

8451 
8156 
7869 

8421 
8127 
7841 

5  I0  '5 
5  I0  '5 
5  I0  J4 

2O  25 
20  24 
19  24 

52 
53 
54 

7813 
7536 
•7265 

7785 
7508 

7239 

7757 
7481 
7212 

7729 
7454 
7186 

7701 
7427 

7i59 

7673 
7400 

7133 

7646 

7373 
7107 

7618 
7346 
7080 

7590 
73i9 
7054 

7563 
7292 
7028 

5  9  H 
5  9  H 
4  9  13 

18  23 
18  23 
18  22 

55 

•7002 

6976 

6950 

6924 

6899 

6873 

6847 

6822 

6796 

6771 

4  9  13 

17  21 

56 
57 
58 

•6745 
•6494 
•6249 

6720 
6469 
6224 

6694 

6445 
6200 

6669 
6420 
6176 

6644 

6395 
6152 

6619 

6371 
6128 

6594 
6346 
6104 

6569 
6322 
6080 

6544 
6297 
6056 

6519 
6273 
6032 

4  8  13 
4  8  12 
4  8  12 

17  21 

16  20 
16  20 

59 
60 
61 

•6009 
'5774 
'5543 

5985 
5750 
5520 

596i 

5727 
5498 

5938 
5704 
5475 

59H 
5681 

5452 

5890 
5658 
5430 

5867 
5635 
5407 

5844 
5612 

5384 

5820 

5589 
5362 

5797 
5566 

5340 

4  8  12 
4  8  12 
4  8  ii 

16  20 

15  19 
15  19 

62 
63 
64 

•5317 
'5°95 
•4877 

5295 
5°73 
4856 

5272 
5°5i 
4834 

5250 
5029 

4813 

5228 
5008 
479i 

5206 
4986 
477° 

5184 
4964 
4748 

5161 
4942 
4727 

5139 
4921 
4706 

5"7 

4899 
4684 

4  7  ii 
4  7  ii 

4  7  ii 

15  18 
15  18 

14  18 

65 

~66~ 
67 
68 

•4663 

4642 

4621 

4599 

4578 

4557 

4536 

45'5 

4494 

4473 

4  7  10 

14  18 

•4452 

•4245 
•4040 

443i 
4224 
4020 

4411 

4204 
4000 

4390 
4183 
3979 

4369 
4163 

3959 

4348 
4142 

3939 

4327 
4122 

3919 

4307 
4101 

3899 

4286 
4081 
3879 

4265 
4061 
3859 

3  7  10 
3  7  10 
3  7  10 

14  17 
14  17 
13  17 

69 
70 
71 

•3839 
•3640 

'3443 

•3249 
•3057 
•2867 

3819 
3620 

3424 

3799 
3600 

3404 

3779 
358i 
3385 

3759 
356i 
3365 

3739 
354i 
3346 

3719 
3522 
3327 

3699 
35°2 
3307 

3679 
3482 
3288 

3659 
3463 
3269 

3  7  10 
3  6  10 
3  6  10 

13  17 
13  17 
13  16 

72 
73 

74 

323o 
3038 
2849 

3211 
3019 
2830 

3l9l 
3000 
2811 

3172 
2981 
2792 

3153 
2962 

2773 

3134 
2943 
2754 

3H5 

2924 
2736 

3096 
2905 

2717 

3076 
2886 
2698 

3  6  10 

369 
369 

13  16 
13  16 
13  16 

75 

•2679 

2661 

2642 

2623 

2605 

2586 

2568 

2549 

2530 

2512 

369 

12  16 

76 

77 
78 

•2493 
•2309 
•2126 

2475 
2290 
2107 

2456 
2272 
2089 

2438 
2254 
2071 

2419 
2235 
2053 

2401 
2217 
2035 

2382 
2199 
2016 

2364 
2180 
1998 

2345 
2162 
1980 

2327 
2144 
1962 

369 
369 
369 

12  I5 
12  I5 
12  15 

79 
80 
81 

•1944 
•i763 
•1584 

1926 

38 

1908 
1727 
1548 

1890 
1709 
1530 

1871 
1691 
1512 

1853 
1673 
H95 

1835 
l655 
H77 

1817 
1638 
H59 

1799 
1620 

1441 

1781 

1602 
1423 

369 
369 
369 

12  I5 
12  I5 
12  I5 

82 
83 
84 

•1405 
•1228 
•1051 

1388 

I2IO 
1033 

1370 
1192 
1016 

1352 

"75 
0998 

1334 
0981 

1317 
"39 
0963 

1299 

1122 
0945 

1281 
1104 
0928 

1263 
1086 
0910 

1246 
1069 
0892 

369 
369 
369 

12  I5 
12  I5 
12  I5 

85 

•0875 

0857 

0840 

0822 

0805 

fZ!Z. 
0612 

0437 
0262 

0769 

0752 

0734 

0717 

369 

12  I5 

86 
87 
88 

•0699 
•0524 
•0349 

0682 
0507 
°332 

0664 
0489 
0314 

0647 
0472 
0297 

0629 

°454 
0279 

0594 
0419 
0244 

°577 
0402 
0227 

°559 
0384 
0209 

0542 
0367 
0192 

369 

369 
369 

12  I5 
12  I5 
12  I5 

89 

•0175 

0157  j  0140 

0122 

0105 

0087 

0070 

0052 

0035 

0017 

369 

12  14 

N,B.  —  Numbers  in  difference-columns  to  be  subtracted,  not  added. 


VOL.  I  —  U 


INDEX   TO   VOLUME   I. 


Absorbing  power  for  heat  radiation,  119; 
for  light  radiation,  277. 

Acceleration  of  gravity  by  Atwood's  ma- 
chine, 53;  by  free  fall,  55;  by  physical 
pendulum,  67 ;  by  Kater's  pendulum,  69 ; 
value  of,  at  Cornell  Laboratory,  69. 

Air,  expansion  of,  99. 

Air  displacement,  correction  for,  82. 

Ampere,  definition  of,  155. 

Atwood's  machine,  laws  of  uniformly  ac- 
celerated motion,  50 ;  gravity  by,  53. 

Ballistic  galvanometer,  221 ;    constant    of, 

221. 

Barometer,   cistern,  96;    siphon,  96;  com- 
parison of,  96. 
Barometric  height  corresponding  to  boiling 

points  of  water,  102. 
Battery  resistance,  measurement  by  Ohm's 

method,  214;    Mance's    method,    217; 

half   deflection    method,   192;     Beetz's 

method,  197. 
Beetz's  method  for  measuring  E.  M.  F.  and 

resistance  of  a  battery,  197. 
Boiling  point  under  different  pressures  near 

one  atmosphere,  102. 
"  Bound  "  electricity,  123. 
Boyle's  law,  example,  10 ;  verification  of,  93. 
Bunsen  photometer,  274. 

Calibration,  of  a  thermometer  tube,  30;  of 
a  hydrometer,  88  ;  of  a  galvanometer, 
174 ;  of  a  prism  for  wave-lengths,  272. 

Calorimetry,  107. 

Candle-power,  by  3unsen  photometer, 
274. 

Capacity,  definition  of,  227 ;  comparison  of, 
227 ;  measurement  in  absolute  measure, 
230. 

Cathetometer,  adjustment  of,  29. 


Cell,  standard  Daniell,  173 ;  E.  M.  F.  of,  by 
Ohm's  method,  189;  by  comparison, 
187 ;  resistance  (see  Battery  Resistance) . 

Collimator,  268. 

Commutator,  158. 

Computations,  5. 

Concave  mirror,  focal  length  of,  260. 

Condenser,  principle  of,  130 ;  variable,  135 ; 
capacity  of,  228 ;  arranged  in  series  and 
in  multiple,  229. 

Conditions,  choice  of,  4,  20. 

Conductivity,  electric,  204. 

Conjugate  foci,  for  convex  mirror,  258 ;  for 
concave  mirror,  260;  for  convex  lens, 
261. 

Constant  of  galvanometer,  true,  156 ;  work- 
ing, 156;  by  copper  voltameter,  167;  of 
sensitive  galvanometer,  170;  of  ballistic 
galvanometer,  221. 

Convex  lens,  radius  of  curvature,  257 ;  focal 
length,  261. 

Copper  voltameter,  167. 

Current  of  electricity,  definition,  153 ;  pro- 
portional to  magnetic  field,  153 ;  absolute 
unit,  155  ;  practical  unit,  155  ;  measured 
by  electrolysis,  155,  166;  measurement 
of,  177. 

Curvature,  radius  of,  by  spherometer,  26; 
by  reflection,  257;  of  concave  mirror, 
260. 

Curves,  plotting  of,  8,  21. 

Damping,  66;  of  galvanometer  needle,  161, 
223  ;  theory  of,  223  ;  ratio  of,  224. 

Daniell  cell,  standard,  173. 

Decrement,  logarithmic,  223. 

Density,  from  mass  and  dimensions,  34; 
definition,  79;  with  corrections  for  tem- 
perature and  air  displacement,  82;  by 
specific  gravity  bottle,  81 ;  of  liquid,  81, 


291 


2Q2 


INDEX. 


86,  88  ;  of  salt  solution,  87  ;  of  water,  87 ; 

by  Hare's  method,  92. 
Deviation,  minimum,  of  light  through  prism, 

269. 
Difference  of  potential,  181 ;  definition,  181 ; 

electromagnetic  unit,  186;  at  terminals 

of  battery,  191. 
Diffraction,  273. 

Dip  of  earth's  magnetic  field,  236. 
Distribution  of  "free"  magnetism,  151. 
Dividing  engine,  31. 

Earth  inductor,  236. 

Efficiency,  of  wheel  and  axle,  46 ;  of  system 
of  pulleys,  49 ;  curve,  48. 

Elasticity,  74. 

Electrical  Congress,  definition  of  ampere, 
155;  definition  of  ohm,  205. 

Electrical  machine,  133. 

Electrical  quantity,  220. 

Electricity  "  bound  "  and  "  free,"  123. 

Electrification,  energy  of,  122. 

Electrolysis,  measurement  of  current  by, 
155,  166. 

Electrolytes,  resistance  of,  218. 

Electromagnetic  induction,  232. 

Electromagnetic,  unit  of  current,  155 ;  unit 
of  E.  M.  F.  and  potential  difference,  186. 

Electromotive  force,  181 ;  of  an  electrical 
machine,  135 ;  definition,  182 ;  compari- 
son of,  187 ;  Ohm's  method  of  measur- 
ing, 189;  Beetz's  method  of  measuring, 
197 ;  of  a  thermo-element,  201 ;  of  in- 
duced currents,  223. 

Electroscope,  128. 

Equipotential  lines  in  liquid  conductor,  199 ; 
how  to  construct,  125. 

Errors,  sources  of,  13;  accidental,  14; 
probable,  15  ;  constant,  17  ;  influence  of, 
18 ;  relative,  20. 

Expansion,  coefficient  for  air,  99. 

Fahrenheit's  hydrometer,  86. 

Fall  of  potential,  in  wire-carrying  current, 
194;  method  of  measuring  resistance, 
208. 

Field  of  force,  electrical,  how  to  map,  125 ; 
magnetic,  definition,  138  ;  due  to  a  mag- 
net computation  of,  146 ;  due  to  a  cur- 
rent, 154. 

Focal  length,  of  a  concave  mirror,  260 ;  of 
a  convex  lens,  261. 


Forces,  parallelogram  of,  42 ;  parallel,  42. 
Franklin's  method  for  magnifying  power  of 

microscope,  266. 

"  Free,"  electricity,  123 ;  magnetism,  151. 
Friction,  coefficient  of,  8,  44. 
Fusion  of  ice,  heat  of,  117. 

Galvanometer,  tangent  for  what  angle  most 
sensitive,  20;  definition  of,  153-4;  true 
constant  of,  156 ;  reduction  factor,  157 ; 
working  constant,  157;  to  set,  157;  de- 
flections, how  measured,  159;  most 
suitable  number  of  turns,  162;  best  re- 
sistance of,  162;  law  of  tangent,  163; 
determination  of  constant,  163, 169 ;  sen- 
sitive, 162;  constant  of  sensitive,  170; 
potential  galvanometer,  192;  ballistic 
galvanometer,  221. 

Gases,  properties  of,  93. 

Graphical  representation  of  results,  8. 

Grating,  diffraction,  273. 

Gravity,  by  Atwood's  machine,  53  ;  by  free 
fall,  55 ;  by  physical  pendulum,  67 ;  by 
Kater's  pendulum,  69 ;  for  Cornell  labo- 
ratory, 69. 

Hare's  method  for  determining  density,  92. 

Harmonic  scale  for  hydrometer,  89. 

Henry,  unit  of  induction,  243. 

Heat,  107;  of  fusion  of  ice,  117;  of  vaporiza- 
tion of  water,  113;  specific,  117. 

Holtz  machine,  133;  experiments  with,  134. 

Hydrometer,  Nicholson's,  85  ;  Fahrenheit's 
86;  of  variable  immersion,  88,  91. 

Index  of  refraction,  268. 
Induced  currents,  direction  of,  233. 
Induction,   electrostatic,   128 ;    electromag- 
netic, 232 ;  self,  209-236 ;  mutual,  240. 
Internal  resistance  of  batteries,  214-217. 
Interference  of  sound  waves,  245. 

Jolly  balance,  84. 

Konig's  apparatus,  245. 
Kundt's   method  for  velocity  of  sound  in 
brass,  249. 

Laplace's  law,  154. 

Latent  heat  of  steam,  113;  of  water,  117. 
Least  squares,  method  of,  21. 
Lens,  curvature  of,  by  spherometer,  26;  by 
reflection,  257;  focal  length  of,  261. 


INDEX. 


293 


Leyden  jar,  132. 

Lines  of  equal  potential,  199. 

Lines  of  force  electrical,  123,  125,  127;  how 
to  determine  direction,  127 ;  magnetic 
lines,  definition,  138 ;  positive  direction 
of,  139 ;  study  of,  141 ;  around  wire 
carrying  current,  154;  of  a  permanent 
magnet,  238. 

Logarithmic  decrement  of  magnetometer 
needle,  223 ;  definition  of,  224. 

Magnet  pole,  definition,  139. 

Magnet  field,  definition,  138;  lines  of  force 
in,  141;  measurement  of,  149,  238. 

Magnetic  moment,  140 ;  by  oscillations,  143 ; 
by  magnetometer,  145. 

Magnetism,  138. 

Magnetization,  lines  of,  138. 

Magnetometer,  145. 

Magnifying  power,  of  a  telescope,  264 ;  of 
a  microscope,  265. 

Map,  of  an  electrostatic  field,  126 ;  of  mag- 
netic field,  142;  of  field  around  a  cur- 
rent, 154. 

Mance's  method  of  measuring  resistance 
of  a  battery,  217. 

Manometric  capsule,  245. 

Middle  elongation,  36. 

Modulus  of  elasticity,  74. 

Moler's  method  of  studying  vibrations,  252. 

Moment  of  inertia,  57;  for  parallel  axes, 
58  ;  of  a  thin  rod,  59 ;  of  a  cylinder,  60 ; 
of  a  circular  lamina,  60;  measurement 
of,  74. 

Moment  of  momentum,  57,  222. 

Moment  of  torsion,  76,  144. 

Moments,  principle  of,  42. 

Mutual  induction,  coefficient  of,  243. 

Newton's  law  of  cooling,  in. 
Nicholson's  hydrometer,  85. 

Observations,  3 ;  record  of,  2. 

Ohm,  definition  of,  205. 

Ohm's  law,  182;  method  of  measuring 
E.  M.  F.,  189 ;  of  measuring  resistance 
of  battery,  214. 

Open-eye  method  of  determining  magnify- 
ing power,  265. 

Parallel  forces,  42. 
Parallelogram  of  forces,  40. 


Pendulum,  physical,  67;  Kater's,  69;  uni- 
form bar,  72. 

Periodic  motion,  time  of,  36. 

Permeability,  magnetic,  138. 

Photometer,  Bunsen,  274. 

Pitch  measured  by  syren,  245. 

Polarization,  effect  upon  current,  179. 

Potential,  electrostatic,  124. 

Potential  difference,  181 ;  definition  of,  181 ; 
electromagnetic  unit  of,  186;  practical 
unit  of,  186 ;  at  terminals  of  cell,  191. 

Potential  galvanometer,  192,  195. 

Principle  of  moments,  42. 

Proof  plane,  128. 

Pulleys,  system  of,  49. 

Quantity  of  electricity,  220 ;  produced  by 
induction,  235. 

Radiating  and  absorbing  power  for  heat,  119. 

Radiation  constant  of  a  calorimeter,  no. 

Ratio  of  damping,  224. 

Refraction,  index  of,  268. 

Regulating  magnet,  156,  174. 

Reports,  u. 

Residual  charge,  132. 

Resistance,  definition,  204;  absolute  unit, 
204 ;  practical  unit,  205 ;  coils,  205 ; 
measurement  of,  by  Wheatstone's  bridge, 
206 ;  by  fall  of  potential  method,  208 ; 
specific,  211 ;  temperature  coefficient  of, 
212 ;  of  battery,  214,  217 ;  of  electrolytes, 
218. 

Resonance  of  air  columns,  247. 

Reversing  key,  158. 

Rheostat,  205. 

Self-induction,  209;  definition,  236. 

Sensitive  galvanometer,  162;  constant  of, 
170. 

Shunt  for  galvanometer,  169 ;  theory  of,  175. 

Simple  harmonic  motion,  61 ;  of  transla- 
tion, 62 ;  of  rotation,  64 ;  examples,  66. 

Sonometer,  250. 

Sound,  244. 

Specific  gravity,  79 ;  by  weighing  in  water, 
80;  by  specific  gravity  bottle,  81 ;  of 
liquid,  81 ;  by  Jolly  balance,  84 ;  by  Nich- 
olson's hydrometer,  85  (see  Density). 

Specific  heat,  117. 

Specific  resistance,  211 ;  measurement  of, 
211 ;  of  liquid,  219. 


294 


INDEX. 


Spectra,  of  metals,  271. 
Spectrometer,  268,  271. 
Spectroscope,  271. 
Spherometer,  26. 
Standard  cell,  Daniell,  173,  178. 
Static,  electricity,  122;  induction,  128. 
Strings,  laws  of  vibrating,  250. 
Syren,  245. 

Tangent  galvanometer,  most  sensitive  de- 
flection, 20 ;  law  of,  163. 

Telescope  and  scale,  159,  161 ;  magnifying 
power  of,  264. 

Temperature,  errors  in  determining,  107; 
coefficient  for  resistance,  212. 

Tenths,  estimation  of,  4. 

Thermo-element,  E.  M.F.  of,  201. 

Thermometer,  calibration  of,  30 ;  compari- 
son of,  108. 

Torsion,  moment  of,  76. 

Transverse  vibration,  study  of,  252. 


True  constant  of  a  galvanometer,  156. 

Uniformly  accelerated  motion,  50. 
Units,  6. 

Vaporization,  heat  of,  113. 
Velocity  of  sound,  in  air,  247 ;  in  brass,  249. 
Vibrating  strings,  laws  of,  250,  252. 
Vienna  method  of  measuring  current,  178. 
Volt,  definition,  187. 

Voltameter,  silver,  specifications  for,  155; 
copper,  167 ;  spiral  coil,  167. 

Wave-length,  measurement  of,'  245;  of 
sodium  light,  274. 

Water,  battery,  228;  equivalent  of  calor- 
imeter, 109. 

Weight,  6-7 ;  in  taking  an  average,  17. 

Wheatstone  bridge,  206. 

Wheel  and  axle,  46. 

Young's  modulus,  74, 


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